I think a lot of different elements are in play here. For starters, why do cracks form in the first place? Some sort of shifting, settling, or upheaval occurred. Any substantial solid pathway exposed to the elements is generally broken up into chunks so they break at the predetermined seams like graham crackers. Much like graham crackers, if the seams aren't deep enough you still end up with jagged cracks.
We can see in the pic there is a crack with no seam at an intersection between the road and the driveway. There is always going to be movement at that point. You can either clean it up into a uniform seam with a flexible buffer such as a synthetic rubber or plants, try to fix it with something inflexible which will either fail or move the failure point to something adjacent, or leave it as is.
I highly doubt the grass in the cracks are the cause of any of the major problems with the current cracks. A lack of seams, possible larger roots, weight, erosion, or heating and cooling are all factors that seem to be much stronger than the roots of many smaller plants. You could try looking at
local buildings that are older and have seams with stuff growing in them. If they still look uniform after many years then they are probably a good local plant worth trying out. I would stick to that method vs buying something that may not be well suited to my climate.
I would mostly concern myself with any
trees that may be close by. If roots are causing the upheaval they They will likely continue to do so, and if the tree dies for whatever reason, that
root may slowly rot and leave a void. Ideally you would heavily prune back the tree and keep it cut back to discourage it from growing excessive roots that could cause further damage.
Every time I've fixed a crack like that I put two lines on the ground that are parallel and lined up with something (In this case the road) so the entire crack fits inside, and then grind away at it with a masonry blade until I have a large V-shaped groove cut. Most people don't want any 'weeds' and elastomeric caulk works well. You could easily add sand and whatever suitable plants into the groove as the transition. Scoop it out every couple years, do any touch up grinder work needed to keep the transition lined up, and replant. If anything has roots that are too aggressive, you can remove it and replace it before it becomes problematic. You could also just sweep it clean and chisel off the high points, but I find most people like the uniform look of a consistent seam so I usually recommend spending the time to cut a groove into it.