Be careful putting force on a drill chuck. They were designed for on axis force, if you push on the side it damages the chuck. If it's your spare drill the damage itself may not matter but flying material is not a good thing.
Speaking from my experience as a bearing representative for a major bearing company, I feel qualified to speak on this topic.
I concur, up to a point, but....... Most drill presses have ball bearings in them. The bearings are designed to handle about 10% of the load in the radial (side load) direction than they can handle in the axial direction. A drill press "can" be used from the side, but only if you're extremely careful about how much force you use on it. More than 10% and you will drastically reduce the life of those bearings.
Double the speed of a rotating bearing and you halve the life of the bearing in that machine. However, double the load of a bearing and you reduce the bearing life by a factor of 10. So, if the bearings would have lasted 1,000,000 revolutions, doubling the speed leaves you with 1,000,000 revolutions still, but you get there in half the time. Double the load on the machine, and you reduce the number of life time revolutions to 100,000.
As far as the drill coming apart and having flying pieces/parts, that is very unlikely on a drill press. It's more likely to lock up and stop rotating, while possibly burning up the motor, but I don't see it coming apart from overloading it from the side.
Suffice it to say that I've been using my drill press as a sanding drum/thumb notch cutter (folders), milling machine, etc. for the entire twenty years that I've had it, and have never had any damage done to the bearings. I took it apart about 5 years ago and inspected it, and the bearings looked like new. It's not using it from the side that damages it, it's using it with a heavy force from the side that will damage it. Just have an extremely light touch and lots of patience, and you can get away with it!