OMG I am constantly fighting Charlie, he is in every one of my 12 flower beds. The idea I would seed it is horrific Lol. That being said, I too have it in my lawn and it is pretty there. I would just dig up some patches and plant it around the yard like plugs. Lift some of the sod and rough up the soil, plant and water. It is as tough as nails. Give him and inch and he will head out for a mile. If you have flower beds or borders near this Charlie lawn, it helps to put in a hard edging. Brick, stone, plastic edging etc. In beds that have plants 12" or taller, I don't even try to clear the charlie any more. I think of him as green mulch. The bed of low sedums is the only one that is hurt by charlie. If I don't keep after him the sedums are shaded out and smothered by charlie. I mowe over him and he just thickens up so he makes a nice lawn.

Compare to Cymbalaria muralis. https://www.google.com/search?q=cymbalaria+muralis&source=lnms&tbm=isch&sa=X&ved=0ahUKEwia9OCrhInVAhWFQSYKHYySAMgQ_AUICigB&biw=1920&bih=940#tbm=isch&q=cymbalaria+muralis+leaves

Violets and glechoma are not nitrogen fixers. However, both low growing plants will shade the ground, regulate soil moisture somewhat, and scavenge excess nutrients. I dont know if they will eventually compete with the vegetables for water and nutrients but both plants can turn into a nuisance and spread through large areas if left unchecked. Creeping charlie especially, when happy, has the ability to overrun and smother other plants 6" and under with its stolons and vining growth. Unmown violets can start to get up in height in my yard too (6-8"). Eventually any fruit sitting directly on another plant is going to kill that plant and both it and the fruit are going to rot.

For a lawn with Creeping Charlie, you want to use this: https://www.domyown.com/hiyield-triclopyr-ester-p-4543.html It takes a few applications, but my lawn went from like 35% Creeping Charlie\Wild Violet to virtually nothing. Yes, it's a herbicide, but Creeping Charlie is pretty much impossible to control organically. I have never sprayed this on my garden bed though, so I don't know if it would damage garden plants or not (I probably wouldn't try it unless you can spray with pinpoint accuracy), but it works wonders for a lawn!