Here is another photo that might be helpful.

The idea of a "weed" free lawn has been to product of companies that sell materials that are known as environmental poisons. Today many places restrict the application of fertilizers containing Phosphorus because of the high levels found in lakes and ponds that encourage the growth of toxic algae and too much Nitrogen has been known for years to accumulate in the ground water and poison the water we drink, high nitrates. Many of the "weed" killers are known to kill off more then just the "weeds" in the lawn and often the "inert" ingredients in the formulas combine to became more potent poisons than the material that suppresses plant growth.
Many would prefer to buy something to spread around to quickly solve a perceived problem instead of correcting to soil problems that are the real reason they are having problems. Start with a good reliable soil test to be sure the soil conditions are close to what the grass you are trying to grow needs to grow decently, the pH is in range, P, K, Ca, and Mg are close to the needs and in balance, and maybe even dig in with these simple soil tests,
1) Soil test for organic matter. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. For example, a good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drainsâ too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy
to see what else you might do to the soil to grow a good healthy lawn.

The weeds seem to be another variety of grass, so couldn't you just mow them along with the rest of the lawn?

Maybe it"s the picture but looks to me like you had some annual grass growing and it died off...which is normal. Did you use or add a annual seed last year? If you did add some seed check the percentage of annual to perennial.

Why would it not be preferable for it to be green more of the time? Our lawn probably has at least 100 different kinds of plants in it, some with pretty little flowers, and they all mow just fine.