Do Log Backups depend on Differential Backups?
Differential backups do not truncate the transaction log, so, there's no relation between Diff and Log Backups and both could co-exist in parallel without affecting each other.

This is correct understanding. In full recovery, ONLY a transaction log backup will  allow log truncation.

I have always thought that since Full/Diff backups do not affect the chain log, Differential backups were mostly meant to shorten recovery time by reducing the number of log backups to be restored.

Yes, this is correct as well. Infact, when a log shipping is broken (e.g. some process takes a NON COPY_ONLY log backup), the fastest way to rebuild log shipping is to take a differential backup and restore it to secondary and then take a log backup to establish the log chain.

there's indeed a strong dependency between transaction logs and differential backups. In fact, I wouldn't be able to restore everything If I had lost the latest Differential backup

NO. This is not 100% correct. Diff and t-log backups are independent.
Diff will allow you to skip the log backups that you will need to restore. But to do a Point-in-time restore, you will need a log backup (with STOP AT command to do a point-in-time restore).
For more details, refer to my answer - Log backup vs differential backup if latter is broken
sp_RestoreGene will automatically construct the right sequence for you.

What is the reason behind DIFF backup being required to allow log restores in the Restore Sequence #2?

This is the minimal repo that proves that you do not need a diff backup to allow log restores in the restore sequence.