I have 2 Xbox 360 2nd gen with no hard drives can't get either one of them to pick up any...

You need to use an official Xbox hard drive, or you can use a flash drive, but the ladder will still need to be formatted before it can be used

https://support.xbox.com/en-US/help/xbox-360/accessories/using-usb-flash-drive

They are both Xbox 360 console drives.

One is from first gen and the second is from 2nd gen 360s

So, if I were to get the updated Xbox Series X port of a game, either by buying the upgrade or via Smart Delivery, I wouldn’t be able to play it on the external hard drive anymore? Because I’m pretty sure that the Xbox Series X is compatible with accessories
from the current gen, like an Xbox One controller. Wouldn’t that include hard drives, so you would be able to use them for next gen games?

No you can't use an external only

The 360 requires at least one drive to be a system drive. The operating system or at least parts of it need to be stored on this.

A system drive can only be an internal drive or a USB drive formatted as a system drive. USB system drives can only be up to 32GB.

So you basically have 2 choices.

Either buy another official xbox 360 hard drive which is the recommended option

Or use both a 32GB Flash drive formatted as a system drive as well as a USB hard drive.

I have had a 360 from day one bought a few games at first but now only use it to play HD DVDs. I have about 100 which if I had to up date to the Blue Ray version would cost quite a bit. I was looking at the Xbox One to use mainly as a media center but
if I can't use my HD DVD drive with it then I will probably stick with my 360 and either get another one as a backup or plug the HD DVD drive into my computer and rip them to the hard drive.