refactor: deprecate usage of cursor.execute statements in favor of the in class implementation of execute.

[gmcrocetti]

• closes #xxxx (Replace xxxx with the GitHub issue number)
• Tests added and passed if fixing a bug or adding a new feature
• All code checks passed.
• Added type annotations to new arguments/methods/functions.
• Added an entry in the latest doc/source/whatsnew/vX.X.X.rst file if fixing a bug or adding a new feature.

[gmcrocetti]

Hello @WillAyd.
So this is the follow up of #60376.
I updated the code base to use panda's execute implementation as much as possible. I couldn't replace all places and a simple git grep '.exec' -- pandas/io/sql.py will show you that. Anyways what in my opinion is worth mentioning and asking for a review is in the following:

1. SQLTable._execute_insert
2. SQLTable._execute_insert_multi
3. SQLiteTable._execute_insert
4. SQLiteTable._execute_insert_multi

This should be no problem because we can always wrap that execution around a try-except block (as I did in SQLiteTable._execute_insert)

Would you mind taking a look and LMK what you think while in draft ?