Scott, I have never understood why most people believe that drill press bearings aren't made for a side load. They are ball bearings. I just looked up parts for a Grizzly DP. Bearing # 6202ZZ and 6207ZZ. The same bearing series are used in dust collector blowers, planers, jointers, table saw (6203), etc., which operate with the shaft horizontally which creates a side load force.
What isn't designed for a side load on a drill press is the morse taper on the chuck/spindle. That is designed for end loading.
Can't say as I've ever seen a DP where the entire head goes up & down. I like the idea, but I believe I would have made the arbor shaft as large as possible - but it may be that way already since I don't know the capacity of his chuck.
The bearings can take a side load but many drill presses have an internal taper socket. Its that tapered socket, if badly made, will shake the chuck loose.
Oscillating spindle sanders often have tapered sockets for the various spindles. This has been going on for 50+ years. Since drill press sockets are overhead, gravity isn't their friend.
Many drill press chucks are taper fit so when extreme lateral vibration occurs, they can fall out. With the spindle sander, when the spindle feels a little loose, a pop on the top of the spindle settles things right down.