Reveal your location.

Judging from the toy jeep, you have a small kid who plays in the yard. Most groundcovers aren't great for lots of foot traffic. You can limb up your tree/reduce the canopy to let more light in. Some trees take well to being hacked, others don't. Next, look for a shady grass mix. Avoid anything that says it should work on sun and shade. Don't waste time on weedkillers. Just mow whatever grows. If it's green, it's good. You may end up with mostly mowable weeds, but if it makes a decent lawn, it's okay.

When your kids are grown, replace with a groundcover.

https://www.houzz.com/hznb/photos/traditional-perennial-traditional-landscape-vancouver-phvw-vp~693927

Grass just won't grow well in shade. Period. And it does not need to be replaced with a groundcover. Any shade loving plant will work (per Jj J's link)

If you do have a small child needing flat play space, consider expanding the walkway to feed into a larger paved surface. Can easily be converted into a seating or firepit area at a later date.

I am in South Florida

Creeping thyme is definitely not shade tolerant.

Many of the suggestions from Lantern are not appropriate for Florida.

I have my suspicions about Lantern. They've posted long replies on several fora and there's misinformation in them. Like Boxwood is a tall shrub and plants block noise. I'm guessing AI.