Enable no floating promises rule

In a quick discussion, we seems broadly in favour of enabling https://typescript-eslint.io/rules/no-floating-promises/ which mandates that promises returned from functions don't get discarded without being awaited on at some point. This will mean a few fixups to our codebase where we do deliberately not await on functions that are async because they need to do async stuff but we don't care about the result.

No files, tests, or entry points are named. Read the linked TypeScript ESLint no-floating-promises rule, then inspect the lint configuration and run the project's lint checks to find discarded promises; done means the rule is enabled and intentional fire-and-forget calls are handled without changing intended behavior.

Written by the indexing model from the issue text.