Your mileage may vary. Sure you can write thousands of unit tests. That's maybe a good idea, if those tests bring enough value to continue to exist long term.
However, the higher up the integration stack you go, to service layer, APIs, UI the further removed you are from isolation, and the slower, and more prone to flakiness automated integration tests become. Not only that, even if you somehow manage to NEVER have a flaky test, if a company like Google with all its billions of revenue, can reach a point (https://www.youtube.com/watch?v=nyOHJ4GR4iU - GTAC 2013 Keynote: Evolution from Quality Assurance to Test Engineering) can reach a point where they have so MANY tests that they cannot reasonably run all of them often enough due to resource requirements, then you may find yourself in a similar boat as well.
Simply automating all the checking isn't going to cover everything. This doesn't even begin to address usability issue, certain types of security testing, or performance testing, (which may require some automation, but will likely require a smart individual to maintain and interpret the results)