> unlike f.e. which side of P/NP divide the problem is on

Actually the P/NP divide is a similar case in my opinion. In practice a quadratic algorithm is sometimes unacceptably slow and an NP problem can be virtually solved. E.g. SAT problems are routinely solved at scale.

An NP problem can contain subproblems that are not worst case problems.

It's similar to the gap between pushdown automata and Turing machines. You can check if pushdown automata will terminate or not. You can't do it for Turing machines, but this doesn't stop you from running a pushdown automata algorithm on the turning machine with decidable termination.

Asymptotics has been used to validate tons of statistical tools. This is just another tool being validated.

If you have a tool that you don't know works when data increases (n-> infinity), then you shouldn't use it.

So practicaly, I believe it has serious implications.

It's very much necessary but not sufficient. In real life the sample complexity matters a lot too, which is also asymptotics, but a more important one. E.g. how the central limit theorem is far more powerful than the law of large numbers.