Technical debt: `noImplicitAny`

### Description
Working in a TypeScript codebase is fantastic, and it really shines when using strict typing for everything, as you can be fairly confident when refactoring. I've spent full days refactoring strictly-typed TypeScript codebases, and everything worked on the first try as soon as the type-checker had no more errors to report. We should aim towards stricter type-checking to increase confidence when making changes to our codebase.

This issue speficially includes the work needed to set `noImplicitAny` to `true` in `tsconfig.json`. Setting this to `true` currently exposes 261 errors in main, most of which are easy to fix. Implicit `any` is usually caused by failing inferred types, which in turn is caused by overly lax typing on our end.

Also consider [denying explicit `any`](https://github.com/typescript-eslint/typescript-eslint/blob/main/packages/eslint-plugin/docs/rules/no-explicit-any.md), making the project very strict (but hardened).

### Related issues
- #556

Start with tsconfig.json and enable noImplicitAny to reproduce the 261 errors reported in main. Work through the exposed errors across the TypeScript codebase, then run the project's type-checking workflow until noImplicitAny passes; consider the linked #556 context before evaluating explicit any.

Written by the indexing model from the issue text.