Alligators can already outrun a person on land.

They just choose not to. Because they’re lazy and they don’t like running after their prey.

I suspect the same was probably true of whatever it was we killed off back then too.

Flat solid land? Nah, person wins on any distance over a few feet.

Through brush and marsh? Nobody beats the gator.

That’s not true. And another one.

TL;DR;

They can run up to 35mph over short (20-30ft) distances. Then they can run at a slightly slower (though faster than most humans) pace for a sustained period of roughly 100 feet (30ish meters for the non-'mericans).

In every scenario they can and will catch every human on earth except an Olympic sprinter. And even then, it wouldn’t be a comfortable race for them.

The reason they don’t chase you down and eat you on land, is because they’ve evolved to be ambush predators from water transitions to land. They are lazy. Simple as that. And their food comes to them in most cases. Why would they want to spend all those calories just to maybe get a un-tasty human? They conserve energy so they can perform several ambush attempts for prey, rather than one long chase.

In the end, their current approach gets them more calories for the efforts, which is why they’ve evolved that way.

But they can, on paper, absolutely wipe the floor with you in a sprint over solid, flat land. For 100 feet.