Actually, there are (were) heads designed for dual use, recording and playback, although I don't remember a cart manufacturer ever using them. They were fairly popular on moderately priced home reel to reel recorders. Sony, Miranda (house branded as Lafayette) and others used the technique.
As regards all the cart machine play heads I've ever run across, Gary G is correct.
A major problem was/is the record head windings must withstand bias current as well as record current,which requires some compromises in manufactire; which compromises aren't necessarily compatible with good audio performance.
As an aside, Akai built and marketed a line of reel recorders which used a seperate bias head (They were called "CrossField" machines). Mechanically, they were nightmares... but when they were properly aligned, the performance was excellent.