I had a similar problem with a drill guide I bought a few years back. I wanted to be able to just attach a cordless drill with keyless chuck.
This is what I did:
I managed to find a threaded shaft that fitted the thread on the drill guide shaft (I guess it was 3/8” x 24 UNF, because the chuck is). This took some finding and eventually I got it from a company called Gimson Robotics - it was described as a “spare output shaft for GR02 gearmotor” - the top end is just a round 12mm - looking at their website, it is not listed, so I guess is no longer available. [I don’t want to disassemble this (I probably used some threadlock), but thinking about it, the chuck is female threaded and so the upper connection on the drill guide shaft must be female, so the earlier comment that the piece is upside down may well be correct.]
Then I attached a separate chuck to the drill guide (and I attached the base of the guide to a wider base for stability).
I have a drill stand and a super drill press, but this is really useful for all sorts of things that won’t fit on the drill press or stand where a perpendicular hole is important.
The alternative would be to find a spare drill (with matching thread) and just set it up permanently on the guide.
Cheers