Since the vise is already broken, and I have to buy something or repair mine, what do you suggest?
First off what I think you have for a vice is a carpenters vice or what is called a 'Tradesmans" vice. What you should have is a 'Machinists' vice, much more beefier & tougher.
Notice this part in yours vs. the vice I use to have & use.
The machined square section keeps the jaw alignment and pressing forces more uumm aligned. lol It's just a beefier piece of equipment for clamping heavy metal parts other than wood or pipe as in a tradesman situation (think plumber)
I use to just wack the studs out with a proper sized ball peen hammer, not the biggest sized ball peen in the box but one size down from the largest, more or less the everyday hammer you grab for. One heavy first wack then 2 light taps and the stud is out. Same as putting them back in, any more than that will require some pressing forces. Swagged front drums is different and should be dropped off at your friendly machine shop if you want to keep that result.
Now most of what I've had to deal with has only been 2 to 4 years off the assembly line and your (I guess) dealing with 40/50 years of who knows what/who *** monkey garage has had their mitts on this. What I'm saying is that you don't need a lot of pressure to put the stud in, the press fit is only to keep the stud from pushing back when you install the wheel. The press fit is not to keep the stud from spinning during loosening or tightening, that is the job of the splines. What happens is that when someone installs a stud is that they don't 'Feel' for the spline to align and just press or hammer the hell out of it jamming it in for good. Next removal can be a pain.
When you remove the stud get out your glasses & trouble light and check to see how clean & crisp the splines are. If they are all mashed someone f'ed it up.
Now whether the splines are mashed or crisp when you go to install the new stud rotate it with your fingers to feel the spline engage and sit down a little in the spline, then with a few wacks with a proper hammer, again first heavy wack to get it going, then a couple of lighter wacks to finally seat it. Hammering skills take years too learn. I can still remember my instructor in blacksmith class going on about beating on metal whether hot or cold on the anvil with a blacksmith hammer to move the hammer to the side and let it bounce twice before taking another swing. And something about the 'Ring' and that wood handles absorb the shock better. I can't remember to this day why the bounce was important, but it seemed to work.
Now this is all fine and dandy when you have the axle/hub out of the vehicle but when your doing this in the field (think heavy equipment) you need a little pressing help as you can't hammer it in due to it's installed position so some will have washers or a collar and pull the stud in with a nut and air wrench. Or you can use your ball/u-joint portable press for large tough ones. Again get to feel the splines engage before preceding.
With a variety of collars and adapters these presses can be adapted to many other uses.
Examples:
Yes
Yes
NOOOOOO!
Notice sledge hammer!