Not quite so fast (pun intended).
The spheroidized steel will be difficult to austenitize. A lot depends on the chemistry, but if you have Cr and Mo (as in 41XX, 43XX etc...), and you end up with a structure of ferrite and speheroidized carbides, the ferrite isn't going to want to change to austenite until it gets above 1725F, or it absorbs enough carbon from the carbides. But, the carbides are M23C6 with all of the Chrome and Moly being tied up in the carbides, so they aren't going to dissolve very quickly.
The short version is, yes you can harden a spehrodized structure just by austentitizing as, say 1500F, but you will need a much longer holding time than you would with either a normalized or as-rolled structure. Much longer. If you don't hold it long enough, you will not get all of the carbon and alloy into solution with the austenite and, while you will harden, you be effectively be hardening an alloy with a lower carbon content and lower Cr and Mo (since these will still exist in the undissolved carbides). You can avoid this by either an normalizing treatment prior to hardening, or by austenitizing at 1700F and cooling to 1500F prior to quenching (assuming 0.35-0.45C).