So are you saying that the drive center stays in the lathe, but you can’t get the drive center’s spurs to hold the wood? On my lathes, I don’t have to whack the drive center into the wood. All I have to do is tighten the tailstock, and the spurs sink into the wood. Maybe you should get a “Steb Center”, rather than the “spur center” (with the 4 prongs). It is easier to get into the wood, and makes catches less dangerous. Here’s the one I have: https://amzn.to/2YdrdSD
Just make sure you get a steb center with the right “Morse taper” that matches your lathe’s spindle.