It’s actually something that’s available on industrial quality live centers, and with an industrial price to match. https://www.riten.com/products/live-centers/spring-loaded-concentric-live-centers/
However what you seem to be experiencing with your current live center certainly isn’t normal as others have already said. I’m only guessing, but you haven’t always been using far too much screw pressure from the tail stock hand wheel in the past? Forcing any live center with too much pressure can or could ruin the bearings quite quickly. All you need is to just seat the 60 degree live center into the previously center drilled hole in the shaft, then lightly pre load it a slight amount to fully take up any tail stock feed screw backlash, then lock the tail stock quill.
At most, the center might need retracting a tiny fraction if the shaft heats up during longer periods of turning to compensate for it expanding in length. Too much tail stock feed screw pressure is also hard on the lathes head stock bearings as well. I suspect you now need either new bearings for your live center or a replacement if I’m understanding the issue your having correctly.