Update to TypeScript 5.6

[wgoehrig]

⚠️ This is still a work in progress - currently blocked waiting for a fix to api-extractor (which is also blocking typescript-eslint, another prerequisite).

It's TypeScript update time again!

Since our last update, there have been 3 major TypeScript versions released (remember that TypeScript does not follow semver). As always, the release notes are the best way to catch up on what's changed since 5.3:

• TS 5.4 release notes
• TS 5.5 release notes
• TS 5.6 release notes

Even more than usual, these new versions bring better compiler performance and (especially) better type inferencing. Other highlights include:

• Regex Syntax Checking - TypeScript can now tell you when you have syntax errors in your regular expressions.
• Isolated Declarations - We're certainly not ready to opt-in to this new, stricter enforcement for type annotations, but someday this may allow us to fully parallelize the entire monorepo's build!
• Disallowed Nullish and Truthy Checks - Some expressions that always evaluate either truthy or nullish will now be flagged as errors. Very handy to catch these at compile time since these can be easy mistakes to make.
• Iterator Helper Methods - This implements support for a very cool new ECMAScript proposal that will make it much easier to work with generators and iterators.

Most of the breaking compiler changes were due to the newly-disallowed nullish and truthy checks affecting some patterns in testing/debugging code. However, better type inferencing also means some new lint errors, and while I'm going to try to hold off on updating ESLint until #7233 there are of course a few lint failures caused just by updating our TypeScript version.

[aruniverse]

this seems like a big change we need to make sure we are checking against