In a world with a continued Cold War and no nuclear detente wouldn't the US want to have a delivery system about as capable as the Soviet SS-18 Satan and the Chinese DF-5A.
No. If you look at the actual discussions that were taking place around the U.S. nuclear force in the 1980s, they mostly had to do with justifying the continued existence of the ICBM arm (the SLBM and bomber arms were quite secure) and trying to fight what was perceived as the biggest weakness of the force, survivability. It was believed that silo-based missiles would be easily and quickly destroyed during a nuclear war, so to be worth building new ICBMs needed to offer some advantage in survivability over the older designs. Some attempts were made to design silos that would be adequately improved in survivability, but these collapsed either because it was impossible (you start needing impossibly tough silos) or the side effects or costs were politically undesirable (many schemes involved turning enormous amounts of public land into missile bases where missiles would be covertly shuttled between numerous silos, any of which might or might not be empty, which was very unpopular in the states where that public land was found). In the end, the only practical schemes had to do with dispersion, that is being able to move the missiles around such a large area that it was impractical for the Soviets to nuke
all of the missiles before they could launch. Peacekeeper had rail-mobile Peacekeeper, but that was always politically marginal and had major issues in clogging up the U.S. rail network if it actually had to disperse. So, the focus turned towards the smaller, road-mobile Midgetman, which was easier to justify and not as politically fraught.
No one, so far as I can tell, particularly cared about or was interested in matching the Soviets or Chinese in throw weight. I'm not precisely sure why, but I suspect it came from a combination of U.S. technical sophistication (our missiles/bombs were probably more reliable and accurate, so fewer of them were needed to hit a given target), better intelligence (the Soviets at least seemed to be very interested in countering ABM systems that never existed), and perhaps relative disinterest in the ICBM force on the part of Air Force brass (how many missile men have been at the top?). For this reason, I don't think it's likely that a "throw weight race" is likely to produce interest in retaining Peacekeeper. What people will focus on is whether the ICBM system is capable of meeting the desire for a reliable second-strike option, not whether the U.S. can launch as many warheads per missile as the Soviets or Chinese. And that metric is probably better fulfilled by more, smaller missiles than fewer, bigger ones.
An additional factor is that even if the Cold War continues further arms limitation treaties are entirely possible. SALT was a product of the 1970s, after all. It's entirely conceivable that the 1990s or 2000s sees a detente 2.0 that results in the Peacekeeper dying anyway.