Office of Steam Forum for Model & Toy Steam Gas & Hot Air Engines

That is pretty clever Doug, will certainly keep that one in mind !

...I have tried turning thinner needles with a finer taper on the lathe, but they never come out right, or they bend before I'm finished....

I have a friend who is very much into old RC engines, and faced the same challenge a couple of years back. He found that it with tiny tips like this, is much easier to "grind" them instead of cutting.
He uses his Dremel in the lathe, with one of these adapters:
https://www.ebay.com/itm/The-ORIGINAL-Dremel-tool-holder-frame-1-2-in-X-1-2-in-Shank-for-your-lathe/113612709777


Might not be of use, or even interest to you, now that you have found this solution to your immediate problem.
Just thought iŽd share, in case someone should face a similar problem, with something needing a slightly different angle than that of a phonograph needle.