Webpack does not bundle the type checking by default - iirc it cannot handle typescript at all by default, it needs to be configured.
The typescript plugin that you used may have included type checking, but it very much depends on how you set it up. Many common setups use Babel to compile typescript code, which doesn't do any type checking at all, just mechanically strips away the type signatures. I would be very surprised if much of webpack's success has come from the type checking functionality you describe.
As for whether it's "the only workflow that makes sense", I find the most comfortable flow for me is using the compiler as a linter rather than a "true" compiler. It runs in my IDE, potentially as a pre-commit script, and in CI (which means checked in code cannot fail to type check), but I can still build failing code. This would be, as you say strange if I were using C#, but syntactically valid typescript can always be covered to syntactically valid javascript, so the output won't be "broken" in the same way that a C# program with invalid types would be. Most of the time, a program that isn't accepted by TSC is still a valid javascript program, just with poorly specified input and output types.
That said, while it can be useful occasionally if I'm still trying to figure out the types, the advantage here is less that I can now compile invalid code, and more that my bundler doesn't need to do unnecessary type checking. Any errors I can catch myself based on the red squiggles in my editor, so I know not to look at the browser until I've fixed all the issues, and at that point I don't need my bundler to also confirm that the types are accurate. Each tool does one task well. (Similarly, I don't want my bundler to also run all the unit tests before it compiles, even though I would normally expect the tests to be passing before I check the output of my code. 90% of the time, if I run the bundler, my tests are passing, and in the last 10%, I probably want to quickly try something out without changing the tests. The same logic usually applies to my types.)
As a result, I strongly disagree with the claim that your process is the only one that makes sense. It may be the most logical for you and your team, but I tend to find that it makes sense to approach typescript from a different angle to most typical compiled languages, and I've found a lot of success in my process.