Some good points. I'd add a few:
I would check the spindle taper seat directly with a tenths DTI so you are reading that surface itself for runout or eccentricity. Introducing a collect and a pin are potentially 2 more sources of error that may either mask runout or artificially exaggerate it depending on how the tolerances stack. Better than nothing. He makes a good point about bringing your tools & having a plan.
I would give the lubrication system like 1-shot oiler a lookover. If its plugged could be a sign of neglect. If its leaking, replacing the lines & fittings is kind of a deeper disassembly.
Funny he didn't mention the motor or electrics at all. Hopefully we are all aware that a 3-phase motor coming from an industrial shop will require a rotary phase converter or possibly VFD conversion depending on some variables for home based single phase power. I'm no expert but I'm told some of the older Bridgeport motors are a bit more obscure & harder to source or retrofit, so might represent some bucks if worn. Lastly ensure your shop panel & wiring amp duty accommodates the new machine. A that pretty much exhausts my entire reservoir of electrical knowledge LOL
This series shows some of the inner workings of Bridgeport style mills.
https://www.youtube.com/channel/UCiqCbERb5VG_G34vkx7j-zg