Can I use an Xbox One S Seagate hard drive to transfer my games to a Series X when I get it?

If they're Series S/X games or enhanced games, they can only run on the internal hard drive or the Seagate expansion card made for the Series S/X. A regular external drive can be used to store the games if you're not playing them and store/play older games.

Any games that are Series S/X optimized can only run on the internal hard drive or the Seagate Expansion card. The external hard drive you are using can only store them.

Xbox Series X|S games can only be installed on the console's internal storage or the external Seagate SSD card. Link here. External drives are too slow for those games. The external hard drive can only be used to install backward compatible games (Xbox, Xbox 360, Xbox One).

Not really...

Xbox Series X games / upgrades / updates has to be installed onto the Series X internal or expansion card SSD,

You can have Xbox One games on your external hard drive that will get an Series X upgrade but the games will need to be transferred to the Series X and then the Series X upgrade will need to be downloaded onto the Series X SSD,

As an example...

I have Marvel Avengers on my external hard drive that will get the Series X upgrade and I will transfer the game over to the Series X SSD ready to download and update the game to the Series X version when available.

Xbox Series X|S games can not be run from USB storage devices. These drives are too slow for Series X|S games, regardless of them being on Game Pass or not.

Series X|S games can only be played on either the Internal SSD, a Seagate Storage Expansion Card for Xbox Series X|S or the WD_BLACK C50 Expansion Card for Xbox