Blockchain – Technically speaking, how do you recover a transaction using your private key?

I’ve searched for this answer, but I guess I don’t know the right keywords, etc., because I can’t find an explanation.

There are 2^96 addresses and ~865,000,000 transactions per private key.

If I have a private key and want to recover the transaction, I guess I’ll have to scan the ~500GB blockchain to see if my address was used or if the transaction was encrypted with my private key.

Even if I had a blockchain on my local computer, this is an incalculable amount of math.

I don’t even see how that’s possible using wallet software like Electrum that doesn’t store the blockchain locally.

However, if you somehow manage to get your private keys into your wallet software, you can recover all your transactions.

How does it work technically? When I use a wallet like Electrum, do I have to send my private keys to a supercomputer somewhere?