YES you would think that if it is METAL that your Metal Detector would go off

However, I do not know how sensitive your Metal Detector is....or what setting it is on.

You have said:
"The object is magnetic (but not very strong)

-> magnet and object attract each other
-> object and other metal do not attract"

You might try turning up your sensitivity on the Metal Detector and checking a few variables?

Looks like from what I've read so far that MOST Tektites would NOT be Magnetic, and NOT have Nickel and Iron Spherules. The RARE examples that DO contain metal have wide variations in amounts. I guess IF the metal content were high enough, it could produce weak? magnetism....?

So I would think we cannot YET rule out Tektite. Perhaps if you look at the surface ( do you have a magnifying Jewelers Loupe? or Macro Lens?) to see if it looks like glass- as a Tektite would- that could help. Of course wear and other factors will shape a glass surface, but I would think something identifiable could be discovered, especially in the "rough" areas.

More Info on Tektites: https://www.tektites.info

CLOSE MACRO PICS of the ROUGH surface areas could help us understand what the material is- such as surface fracture patterns ?