You can resharpen the tip, just tape the live center so it turns when running the lathe then hold a file to the tip and file it til it's sharp. The live center needs to be in the head of the lathe for this.
That might be the problem. it is from an 80s model lathe. The lathe had barly been used and I had to clean off all the rust. One factor of it being a cheapo might be after turning a while the metal will get hot to the touch.
My live centers do this too. I was told by a machinist that it was because the live center is not coming right up to speed with the lathe so there is some slippage as it speeds up. I use my disk sander to resurface mine. I simply hold it up to the disk and let it rotate and use my thumb to regulate the speed. It only takes a few seconds on the sander to fix it right up!