Use `ts-bridge` as build tool, set `moduleResolution` to `NodeNext`

[MajorLift]

Explanation

As part of our OKR for upgrading TypeScript to v5.0+ (Q2 2024 O3KR4), we are updating our core dependencies to generate builds and type declarations for both CJS and ESM.

References

• Same update made to @metamask/utils: Use ts-bridge for building utils#182
• Required to update core repo's moduleResolution setting to NodeNext:
  • Blocks: Upgrade to TypeScript v5.0 core#3651