Andrew
As you have discovered the common method is to use some sort of ball bearing spindle with a chuck on one end and pulley(s) on the other to take a belt drive. Drive either the old fashioned way via an overhead shaft and long belt or from a close coupled electric motor fixed to the same mount as the spindle.
Which seems an awful faff for something that will, most likely, see little use.
Quick and dirty bodge was to clamp an old style mains powered pistol drill to the toolpost via the standard size collar most breeds have. Effective but somewhat uncontrollable and lining up could be aright pain. I abandoned an attempt early on because I could see no way to reliably set the drill horizontally on centre as a starting point for setting an offset to do drilling on a pitch circle and similar things.
Trouble with attempting to use a battery drill its that generally they don’t have a mounting collar and modification of the plastic case is essentially impossible in most cases. All the bits and bearings being held via cunningly moulded webs, slots and half holes relying on the case being complete and screwed together as the designer intended if it is to retain its structural integrity. If hoped to use one of the Makita NiCad drills made redundant due to switching to Lithium battery versions but there seemed no reliable way of modifying the cases.
As Russell notes Dremel and similar have a suitable collar type carrier moulded into the case but they re limited to drills small enough to fit the collets.
Inline die grinders look promising as most have a mounting collar but they run way too fast with too little torque for drilling.
I concluded that the old fashioned way with a spindle and close coupled motor was the only approach with a high confidence of being successful. Too much faff for too little gain.
If you do go ahead and make a spindle an effective alignment method is to modify an MT blank end arbor with a suitable external diameter or centre hole to fit something on the pulley end of your spindle. Sit it in the tailstock and adjust the cross slide so the mating feature on your spindle slides in or over. Works a treat with the carrier for my tube mount coventry die head, which has to be on centre.
Clive