Hope this isn’t against any rules I am unaware of but I too have been having troubles using the lathe due to the misaligned tailstock, the work around that I have been using to drill holes into turned down stock however is to simply mount a spot drill into one of the tool holders and eyeball dead center on your work and then proceed with drilling as normal. I have planned to use this week to look into how to align the tailstock with the headstcok as well as try to fix the locking mech. for the tailstock as I have a project I am trying to complete by the end of this week that requires the use of the lathe. I’ll post updates here if I make any progress on fixing the equipment.
For now however, using drills in the tool holders has worked for me as a means of drilling holes in my work. Just a suggestion if you really need a hole drilled using the lathe ASAP.