I think/hope that the wording you used was a mistake.
End to end tests do not introduce flakiness, but uncover it.
Whenever we discover flakiness, we try to fix it immediately. When there is no time for the fix (which is more than often the case) we create a ticket that vanishes in the backlog.
For a long time the company I currently work at didn't have end to end tests save unit tests for a lot of their code.
Through a push of newcomers we finally managed to add end to end tests to many more parts of the code. However, these are still not properly documented. Some end to end tests overlap and some only cover a small part of one larger functionality. That is why we often find bugs that were introduced by us, because we had no end to end tests covering those parts.
We used to run end end tests only every night on the whole product. They usually take an hour or more to complete. This takes too long to run them before each merge. However, we have them organized enough such that for sub-product A we can run the sub-product A end to end tests only before each merge where we assume that we did only touch code affecting sub-product A. In case the code changes affected some other parts of the product, the nightly tests help us out. We are doing this in my team for a long while now. But we just recently started to establish this procedure in the other teams of the company, too.