Absolutely you can cold compost as you describe; much of my waste organic matter is placed uncomposted with plants to rot in the growing environment.
Renais

For small batches of kitchen scraps,rabbit poop and other "green" items,that's an exclent plan. Pull mulch back , bury additions in soil then replace mulch. Some go as far as burying large quinities of greens and manure in a ditch between rows or in vacant rows. Same as bins or piles,don't bury things like fish and other items that atract animals.

"Cold composting" in the garden is also known as sheet composting or mulching. Laying organic matter on the soil, as a mulch, will aid in unwanted plant growth control, aid in soil moisture control, aid in keeping soil temperatures more moderate, and add organic matter to the soil, all benefits.
You can also, as Klem suggests, bury that organic matter, sometimes refered to as side dressing, but that may result in a temporary tie up of soil Nitrogen as the Soil Food Web works on digesting that material.

There are several variations of "in-situ" composting for gardens.

Trenching: Make a trench or a series of holes between the rows of veggies and bury your kitchen scraps and weeds as you acquire them ... next year, put the rows where the trench was.

Toss and forget: unless they have seeds, drop the weeds and trimmings onto the dirt between the rows and leave them to decay. Scatter kitchen scraps the same way. Till them in the next spring.

My compost is under constant assault by rats, skunks and crows, so if I were to shallowly bury in the garden it would be dug up that night or sooner.

You need to get a gun or dog pnbrown

Yeah, a varmint rifle for those cows. LOL

Sheet composting is basically how nature does it!

The key to minimize opossum incursions is to start composting in a trash can. Even when they are ravenous they will not eat one month old, moldy cornmeal (n two months there will be no recognizable cornmeal). When you pour the can contents into the trench, it smells really bad, but within two days the smell has abated and within a week, disappeared. In fact, the bottom of the can is an unrecognizable black goo (maybe herb stems are recognizable, but that's it), the middle putrid stuff, and the top still has recognizable materials.

In our case we produce two trash cans of kitchen scraps a year (lots of discarded materials from the garden, e.g. outer leaves of cabbage and collard), that is all the fertilizer for three beds.

Any time compost has a putrid odor there is a problem with the mixture. Compost should never have a disagreeable odor, either sewage like or ammonia like, if it is mixed properly.
My experience is that putrid smelling compost is more attractive to unwanted critters then is compost with that good, rich soil smell it should have from the beginning.

I gave up large outdoor piles because of the attraction of unwanted critters. I could probably have lived with some of them, but the first unwanted critters also attracted rattlesnakes, and there is nothing more stimulating than turning a compost pile with a rattlesnake hanging from the pitch fork. I use some larger compost systems inside the greenhouse where the critters can be controlled. It is true, as has been mentioned, that there is some odor when the larger containers are dumped on the ground, or when one is dumped onto another. My method of dealing with the containers is to dump them one into another (I make huge amounts of compost a year, so I do this when I have the urge only). The dumping causes the material on the bottom which is wet to be on the top where it can dry out a bit. The odor only remains for me for a few hours, and really does not bother me. The odor is just anaerobic composting--bokashi is an example of anaerobic composting. The material will still degrade well, and it will still make a great final product. My containers have only one, 1/4" hole in them, about 2" up from the bottom to drain huge excesses of water. So, if it has been rainy, I can expect to have two inches of water at the bottom of the container. In our dry environment, I appreciate the rain providing some of the water needed for rotting, and I don't mind dumping the containers to redistribute that moisture to the rest of the materials. By the way, I find that garden debris that has sat in these containers for a month or two becomes quite hydrophilic, and easily absorbs the water on the bottom when turned. I also think I get the compost finished faster with the anaerobic action than I would with just aerobic rotting. I do not shred or grind any of the garden waste--too much work-- and the wet rotting takes care of breaking up the big waste for me.

Renais

I'm pretty relaxed about critters in the compost, but I have to agree with you, I'd draw the line at rattlers!

Our food scrap compost at work (office building) gets transferred from plastic bins to an open bin to finish, and we tarp it to keep heavy rains out. Sometimes in early spring there will be a big black snake under the tarp keeping warm and looking for mice. A harmless and beautiful creature. I don't tell the other volunteers about him, and he doesn't bite, so we're good. :-]

kim, I don't compost in the trash can. I degrade edibility.

Having problems with offensive odors in a composting operation says the person building that pile is not doing something correctly. Perhaps spending some time with this Composting Tutorial could help.

Here is a link that might be useful: Composting Tutorial

Although I'm not in favor of stinky compost, if nature wants to do it anaerobically, I'm not sure we can say that's 'incorrect.' :-] It's just not our conventional preferred way.

All I do is till my garden every fall (3,600 sq. ft.) and add about 120-150 large bags of dry leaves that a half dozen neighbors throw next to my fence, I spread them out and let them finish drying for a few days and I roto till them 12 in. deep going both directions. I do this at the end of Sept and with all of our rain and snow by the next April I till it once more and seed/plant. I have done this for at least 10 years and it has really helped my clay soil. I do however broadcast 10 lb. of 20-20-20 and with proper crop rotation have excellent vegetables and potatoes. I have found that horse manure contain way to many seeds even after piling it up for a year which is a pain in the ass, because cows better digestive system that's what I use.

You can make compost anaerobically, in a digester and capture the gases produced for use in cooking, heating, or electricity generation, but that is a very closed system and would not include a container with numerous holes bored in it.
For those interested J. I. Rodale published a lot of information about anaerobic composting in the 1970's that did little until of late and now people are "discovering" the process that has been around since about the late 1800's.

Here is a link that might be useful: About anaerobic composting

robb61: I guess that's not much fertilizer but just watch out for K and P buildup in that clay over time. I added compost and very little fertilizer for 20 years and have ended up with high K and P to the point where I'm slacking off on even the compost additions and not using any fertilizer at all. You might get a soil test if you haven't done one for years. Just a helpful bit from another clay gardener. :-]

I have used to dig in kitchen scraps in veggie beds until I got rats digging after me.
Now I do bucket composting of kitchen scraps, I have several buckets with holes in bottom. Add scraps, close lid, make small hole in garden bed and put bucket there.
Worms and addition of water from rinsing buttermilk glasses and jars give me a great compost that I mix with compost from the pile where I put garden scraps and UCG.

If critters become a problem digging up buried vegetative waste that should tell you that material was not buried deep enough, or too much material was put in that area. Care in burying kitchen waste is necessary so pollution does not become a problem as it is in the landfills where most all of the household waste is dumped.

Kinda straying off subject but in response to Tox's comment regarding higher than desired levels of P&K. I once had soil with especialy high P that I planted to Yuchi Arrowleaf Clover. The clover uses a lot of P and K while fixing up to 200 Lbs N per acre. In haying tests,yields run from 4 to 6 tons per acre (hello green manure). Year 1 and 2 I turned plants under in april leaving narrow rows to mature and seed in June. I planted 2 rows of okra between single rows of clover. The clover was cut in late June with side discharge mower which in turn mulched okra. Okra was mowed then all rows lightly disked in early October to cover clover seed. Soil tests in 3rd year spring showed normal P and K plus high N so all plants were turned under and normal ground use resumed. I still have volinteer plants every year in out of the way corners. I find these plants handy to mow for green manure starting in March. I realize many people are limited on space and/or ordinance against high weeds but thought those with space might find it interesting. Besides,I thought it might be a refreshing break from B+G ratio conversation. It's difficult finding seed in less than 50 Lb bags but a few minutes in June on a roadside or farmer's pasture will yield enough to plant several hundred sq ft of ground in fall. With the exception of horses who will not touch it,livestock love the stuff. Wildlife lovers should be aware deer,turkey,rabbits and birds are attracted. Not so sure gardeners want to hear about wildlife attraction.