Some live oaks put up lots of suckers. Some don't. We actually have two species.
is the usual yard tree. Fla. plant atlas

Sand live oak has leaves that are more curled at the edges and fuzzier underneath.

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It's usually a scrubby tree, forming thickets and adapted to periodically being burned to the ground.

Both species can make suckers. And of course both have lots & lots of small, shallow roots forming a sort of sponge under the tree. In my yard, St Augustine grass does pretty well under laurel oaks, but hardly anything else does, except border grass (Liriope) or shade-loving bromeliads.

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