Traveling Salesman is NP-Hard, yet Uber Eats delivery route optimization algorithms exist

Actual difficult instances of TSP are pretty rare, and for something like Uber Eats, it's fine if your route is 2% worse than the mathematical optimum. Traffic fluctuations probably matter more than having the shortest route.

There are many good heuristics for TSP that might not give you the optimal solution, but that will generally come pretty close. The Wikipedia article probably describes some of these.

NP just means the algorithm to find the solution takes longer to run than verifying a solution is correct. If verifying the solution is easy then shouldn't there be a way to find the solution easily? That's the P=NP problem. NP-Hard does not mean the problem is hard.

Algorithms that find approximate solutions to Traveling Businessman Problem are handful (some just use Markov Chains, a rather easy topic). Finding the exact solution is a hell lot harder.

If your solution has an estimated error margin of 2% or less, it works just fine for basically any practical purpose.

Uber Eats has it easy. It is a problem for the USPS and delivery companies, including Amazon.