As long asyou keep them from choking out your other plants, it should be fine. The strawberry will also spread into the lawn areas though.

I think they are pretty invasive but easy to destroy. My daughter has strawberry hybrids called 'Lipstick' and 'Pink Panda' that would romp across the landscape wherever there is sun, if she turned her back on them. She has to re-direct the runners back into the area every week or so.

i've never seen the choke out healthy plants, myself- certainly not anything taller than they are!

but I have dense borders where only the strong survive- I can see some of the more tempermental plants (roses come to mind) not liking any sort of groundcover at all.

thankfull, they're easy to rip out, in clumps if need be.

that ajuga-looking this is either just another ajuga, or something we used to call wood sorrel (if the flower spikes are sour, it's sorrel) they are matt forming, and somewhat competative- but again, being shaded by taller plants slows them down a fair bit.