Would this have resulted in something like otl's 25th Amendment being passed much sooner?
Well, two subsequent deaths (Harding and FDR) did not have this effect. And the VPcy had been vacant eleven times before without producing such a move. Assuming Hay's presidency went without major incident, such a change might become less likely rather thna more.
Also, iirc, the 25th Amendment was as much to deal with presidential disability as with succession. That wasn't much of an issue in 1901