If grass isn't growing well enough to crowd out those wild strawberries, you might see that your soil is acidic & infertile. Not to suggest you go broke fighting this, but some lawn lime and fertilizer would surely help you to build better turf out there to compete. Wild strawberries will grow on a sand dune, and are prominent on the less travelled gravel roads & ranges on my ten.
Remember also that plants won't absorb 2-4,D or others well unless 'actively growing' as in after a period of rain. A wee bit of liquid fertilizer will stimulate absorption, but now might be the worst time of year to expect good results. Too strong a mix can burn grasses when drought-stressed so know when to say when, and do your work efficiently in months with an 'R' in them. (like when we're transplanting)
Weed-B-Gon may have backed off on the 2,4-D when they added Dicamba. IIRC it wasn't there years ago, but Spectracide has given Ortho serious competition in the last '20'.
Oh, btw, I'd never mow 'short' for any reason. There's just no decent Northern turf grass that will grow at mowing heights so short that there's no weed that WON'T. Does that make sense? Dandelion, plantain, clover, medic, spurge, purslane, cinqefoil, oxalis, yarrow, red sorrel, clover, creeping charlie .. will all thrive even if you mow to 1 1/2". Most fescues, ryes, and bluegrass do best at 3-4". (Go measure that thick green stuff in front of the bank. The color results from irrigation and nitrogen, but density & choking out weeds depends on the right cutting height for a particular lawn specie.