As someone who was involved in politics at the time, there are several misconceptions about the legislative process. First, while the Senate was Republican until 1987, the House was held by large majorities by Democrats. William J. Hughes (NJ) was the chair of the House Judiciary Subcmte on Crime. and was a political ally of House Judiciary Chair Peter Rodino (NJ). The House Judiciary nor the Dem Leadership of the House was not going to budge on the Hughes Amendment in conference on the bill as both Hughes and Rodino were anti-gun as was House Speaker Tip O Neill (close friend of the Kennedys). The NRA was blindsided but felt that FOPA's benefits outweighed the costs and that the rare opportunity for a new law rolling back some older issues was more important than restricting new machine gun manufacturing. Some here disagree, others don't.
What is true is that if FOPA had not been signed, there would have probably been no law passed. GHWB Bush was not pro gun and faced a Dem majority in both houses anyway. Bill Clinton was not pro-gun nor really was GWB (remember he supported reinstatement of the AWB when it expired). While the failure of FOPA would have benefited those wanting fully automatic weapons, it would have hurt others as JohnKSa mentions. That is also assuming that the Dems would not have done something like the Hughes Amendment during the GHWB Bush or Clinton years. It is most probable that either one of those presidents would have signed such legislation.
Remember the genesis of the machine gun law was not FOPA, but the NFA act in the 1930's as modified by the GCA act of 1968. The 1968 Act allowed the executive branch to limit much of the importation of foreign military arms with the requirement of "sporting purposes." GHWB used this to restrict such imports by executive order in his presidency which was later buttressed the 1994 AWB.