As long as the soil media is loose and well draining, the seedlings will produce roots all on their own satisfactory for their proper growth without any root hormone or root stimulator. It is unnecessary and an expense that can be easily avoided.

Here is one discussion that is quite readable: Root Stimulators (Hormones) Along with several more useful suggestions it says

''Plant hormones, known as auxins, can also be used to stimulate root growth. A couple of hormones commonly found in rooting gels are indole butyric acid and naphthylacetic acid. These two hormones are considered root stimulators. Although some growers will use these hormones to stimulate root growth in early vegetative stages, I have only used them to stimulate new roots on cuttings. Auxins are known to promote lateral root development so they are more suitable for cuttings versus seeds."

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When you plant out, with a three-inch tall plant in a four-inch pot, you will be stunned at how many roots you find packed in there. Why in the world would you want more roots? So you need to plant out when the plant is one-inch tall?

"Is that something that is done or that is reasonable to do though?"

No.

"Are there more appropriate ways to help my starts put down more roots?"

Loose, well-draining soil-less mix, proper watering practices, cooler ambience for slower top growth, staged transplanting to trigger new root growth, lo-dose but consistent nutrient levels, etc. all contribute to root development. The size of top growth is not a valid indicator of root development other than if that top growth is exceptionally high then the root growth is likely lagging behind.

Dave

Phosphorus and potassium are both very helpful for roots (both according to what I've read and what I've experienced with potassium—and according to what I've read with phosphorus, but I imagine it's true). Too much nitrogen can be bad for roots, as can too much water, too little ventilation, diseased soil, etc.

When I started plants indoors, I used to give them potassium sulfate, and they would get awesome roots pretty fast. I also did this for cuttings rooted in soil, and for transplants, and it was helpful. I didn't experiment with phosphorus much until I stopped starting seeds indoors.

I'm not saying that potassium and phosphorus do what hormone rooting powder is supposed to do, but at least potassium is still helpful for cuttings rooted in soil (it helps them to absorb more water and be less prone to shock—and at least by the time they do root, it helps the roots). I found it to be extremely helpful (both for seedlings and cuttings, but avoid soils with big chunks of partially composted wood chips). I took loads of cuttings in early 2015 with a very high success rate (e.g. tomatoes, watermelon, muskmelons, cucumbers, peppers, tomatillos, etc.)

If you give cuttings bright light too soon after you take the cutting, it can stunt and/or wither and/or kill them. Give them at least a day or two of dimmer light.

I did pretty large cuttings every time, though (between 3 and 16 inches). Maybe the reason people tend to use younger shoots is to avoid the light issue—but you can definitely take longer cuttings involving older growth. I'd usually take maybe an 8-inch tomato branch, put it in a cup with Nature's Brand seedling mix (with basalt rockdust and some kind of maybe natural 10-10-10 fertilizer added—it was a safe kind for seed-starting and taking cuttings), plant the stem near to the bottom of a 20oz foam cup, and water with water that has a good dose of potassium sulfate in it. Then I'd keep it out of bright light for a couple days, and then put it in bright fluorescent light, or a south window. They generally took a few days (~4 to 8) to root and start growing (for tomatoes; some other things took longer). You don't need hormone rooting powder for the plants I rooted (although you might for some species).

If you believe in rooting things in water, it takes some faith to try rooting in soil, but it really is easier (especially in the long run). It's way faster, too. You can have a nice plant in no-time. Just be careful about damping off—you really need to avoid that (it afflicts both cuttings and seedlings). Fresh soil, clean containers, plenty of light (especially 2700k fluorescent light, for some unknown reason, and brighter south window sunlight), ventilation, and a high soil line can help to avoid damping off disease.

I'd do a few things differently if I were to take cuttings now (I'd probably experiment with using wood ash instead of basalt rockdust and potassium sulfate, and I'd do some judicious pruning of the old leaves before transplanting the rooted cuttings to the garden; I wouldn't add 10-10-10 fertilizer, because I think it caused more mold or something, although the plants did enjoy it). I would add a small fan to my growing area, and maybe use heat mats where helpful (while keeping the air cooler than I had it). I'd probably give more attention to fewer plants, instead of packing loads of them in close proximity. I'd probably use smaller foam cups (and not start my plants so early). The wood ash could help pests/diseases not be as much of a problem.

I used the same soil and amendments for seedlings as I did for cuttings.