I haven't looked at the hightower lathes but I would imagine that the tailstock has a dovetail lock of some sort on it. when it is clamped, that is the only time the tailstock is inline (centered) with the headstock/spindle/chuck. When the clamp is loosened to slide the tailstock there is an amount of clearance there that would multiply by however long the tool (drill) is. Lets say you have a 3" drill in the tailstock and it is loose so it can slide freely like you'd need it so your carriage feed can push it, at this point you wiggle the tailstock and indicate how much play there is, and it's (for easy maths sake) .030". Doesn't seem like much till you throw a 30" gundrill in and that play now becomes .300". Gundrills like to follow the hole there in, but they will walk if there is side load on them. All drills will, and they will take the path of least resistence. If you want to drill with power feed I would probably set up a QCTP (I like dorian, they seem to repeat very well), note all the settings to go back to, and use the carriage. I've seen guys at work try similar things to what you are saying, and there has always been some kind of carnage. Difference being they tied a strap to the carriage and pulled the tailstock. Just my 2cents, thought I'd give my experience and ideas.
Tom