THIS MOST certainly is Japanese Stiltgrass.
There are parallel veins in your plant leaves making it a monocot. Using any herbicide meant to kill broadleafs would have a tough time.
The main thing with broadleaf herbicide is that these plants have BROAD leaves. One is supposed to dampen the lawn before applying these specific herbicides. Broad leaves are able to have more of the herbicide stuck to them than the thin thin bladed grasses. Trimec, is a normal broadleaf herbicid. I don't think it will work well with this 'broadleaf monocot'...
I am thinking that your lawn is a warm season grass. Yes or no? If you had cool season grasses I could easily help you rid yourself. But warm season grasses are extremely different.
Sharpen your blades and COLLECT your clippings!
This weed has a very shallow and weak root system. This will be easy to control because cool season grasses have huge root systems that can be trained to grow very deep.
These grasses with deep roots are able to resource water 4 to 6" below the surface. This weed can only get water 1/2 " to 1" below the surface. I love cool season grasses. I hate warm season grasses because those grasses are similar to the weeds one wants to control.
Cool season grasses, if you learn to manage them correctly make life with a lawn easy.
Quit worrying about 'killing' this weed en mass. Not going to be pretty at all and you will still need to learn how to WEED PROOF your lawn.
Cutting no lower than 3 to 3 1/2 inches. Get your mower deck raised if necessary.
Watering DEEP. Like down to 4" deep in the soil. Then NOT watering until you see your footsteps on the lawn. The blades of grass you bend stay down. THEN and only then do you water deeply again. Your intervals will be short and then lengthen as you train those roots to grow DEEP. 1" of water per week should be what you expect after the roots are trained. Shallow watering only enhances the weeds. Tall top growth prohibits germination of seeds in the lawn bed. Top growth feeds those big root systems and helps to get them to grow DEEPLY. 4 to 6"
Mow NO SHORTER than 3 to 3 1/2". If you have to, get your mower deck manually changed so that you can cut the grass at this height.
Water deeply, cut the grass no shorter than 3" and you better believe you need to fertilize. 4X per season is normal. Dr. Earth's LAWN fertilizer is spectacular, more expensive but you only need 3X application. Got Mycorrhizae fungal spores and best of all thatch eating bacteria! I was responsible for hundreds of acres of lawns to care for per week. This stuff was worth the extra cost!
Using pesticides, which include herbicide/miticide/fungicides...can cause secondary problems. Mowing high, watering deeply and infrequently, fertilizing properly...will cause YOUR crop to out compete the weeds. Gee, I hope this helps.