feat(eslint-plugin): [prefer-nullish-coalescing] add support for assignment expressions

[abrahamguo]

PR Checklist

• [x] Addresses an existing open issue: fixes #10145
• [x] That issue was marked as accepting prs
• [x] Steps in Contributing were taken

Overview

Re-applies #5234, while accounting for updates made since to prefer-nullish-coalescing.

Commits:

• [x] Extracted the current logic of the LogicalExpression listener into checkAssignmentOrLogicalExpression
• [x] Pulled in other changes to rule file, as well as updates to checkAssignmentOrLogicalExpression to support assignments
• [x] ported over test cases and updates to test case generation functions
• [x] changed all declare consts to declare lets — it is not clear to me why @JoshuaKGoldberg did this in #5234, but I have done it