ONE2 mode for pickup machines will allow you to go directly from recording to overdubbing (version 1.25 manual page 56).

There is a gain mode that allows you to vary the gain of the pickup loop, but overdubbing is disabled. It’s been a while since I played with PU machines, but there may be a way to drop the loop when overdubbing.[/quote]
Regarding the gain control (not “gain mode”), this absolutely allows “loop ageing” or what the op calls a feedback control. You can overdub and manipulate this parameter to get the same effect.