The press I have is holding the bit I have fine I have no clue what size it is but measured with my caliper it's .0375 I need to buy a really small set of bit I think because I'm not sure if mine is small enough. This press I've had for quite a long time but like I said never used because I had another and it really has no wobble at all thankfully..
You said you used a #78 bit if you measure that with a caliper what size would it be, I cant read any markings on mine so I'm not sure what size it is but with the caliper it is like I said above .0375, I like the guide idea I'm going to have to work on something like that....