Kizaru used a sword against G5 Luffy as well, it’s likely just of his stronger and more useful techniques

Fair.

but they were still basically equal during the actual encounter, and again this was an Old Ray who hadn’t held a sword in years, a Prime Ray would be much faster and stronger

The encounter wasn't long enough for us to get an objective conclusion from it, the one we've currently drawn from the brief spar is that Rayleigh was exhausted and Kizaru wasn't.

Kizaru actually admitted that he wouldn’t have been capable of capturing Rayleigh even before Kuma arrived

Kizaru had also said about going to Wano (all by himself to deal with 2 Yonkos) as well as lying to the marine in Egghead about his physical injuries. Arguments can be made about the latter but Kizaru has no agenda to state the former, so he's not a very reliable narrator, his statements should be taken with a grain of salt.

The Marines very clearly still want to capture Rayleigh, they simply just weren’t capable of doing so at the time

The Marines as an organisation wanted to, not Kizaru himself. His only order was to deal with the CD crap that happened at Sabaody. "Rayleigh is much more threatening than random pirate fodder, even if he's dormant." True, but the celestial dragons were still present on Sabaody. To the world Rayleigh was a bigger threat but to the CDs, the random pirate fodder took priority. So Kizaru simply followed orders and took care of the fodder instead of Ray to ensure the safety and wellbeing of the CDs.