You can get high precision in any type of bearing, some spindles have precision cylindrical roller bearings in addition to the angular contact.
Tapered roller bearings are used in these lathes because the speeds these run at are relatively low, you can’t run most tapered roller bearings at CNC spindle speeds, you need angular contact bearings for that. TRBs have a higher stiffness than ACBs, so you can get a more rigid spindle system in the same size spindle and housing, that is likely why these machines use these instead of ACBs.
Part of the reason for using high precision in machine tools is so you can have a higher preload than you can with a standard precision bearing. Higher preloads make for a more rigid spindle and give you better performance of the machine. If you think of each rolling element in a bearing as a spring, you can visualize that the more rolling elements in contact with the races (higher precision) the more stiff the bearing will be.
A standard precision bearing might give you good performance, but there is no guarantee, you could get one that meets P5, or you could get one that just meets normal tolerances. Most of us here do not have the resources to check a number of bearings for tight tolerances and send the rest back, so you buy the higher tolerance bearings the bearing manufacturer made. Even though modern manufacturing tech allows for more precise manufacturing, that allows the manufacturer to run faster and not necessarily produce products to a higher precision. Remember, higher precision requires higher testing standards, and that costs money.
FAG makes a bearing line called X-Life that takes advantage of the higher precision of modern manufacturing and can guarantee tighter tolerances for near standard bearing prices. Most of the tolerances meet P6 tolerance class, I don’t recall which, and I’m on a project now and can’t check, but if you can’t find P5 bearings, X-Life would probably be the next best thing, definitely better than standard precision bearings. I’m sure SKF has something similar, maybe check into the Explorer series and see what tolerances those are made to, I don’t recall if they meet P6 or not, but they are supposed to be a higher quality bearing than the standard ones.