finding the weak link

No obvious weak links.

You could try going to a separate pre and power amp.

Your speakers go down to 28hz. You will be surprised at how much more info will be extracted from your main speakers when you ad a subwoofer. Add 2. You won't be disappointed. my speakers are rated to 20hz. yet the subs were probably my biggest improvement.

then your phono preamp is not up to the task. Do NOT underestimate the importance of a GOOD phono preamp. The OL Silver arm is OK and actually hits above its price level as ALL the OL arms do. At some point you might want to upgrade to something like an Illustrious which is supposedly their sweet spot.  I have one that I bought for my Teres TT as a temporary arm. That was 12 yrs ago. I just upgraded to a TriPlanar. Yes it was better. But the OL has absolutely no shame when placed alongside the Tri. keep your eye open. i have it on good authority that there will be one for sale sometime soon. You have a good system in the PL and speakers. i would not consider changing the PL until you take care of the problems I mention.

BTW, like others have said, the KT66 & 6L6 tubes are something to consider. My own favorite is the 7581 tungsol. Fantastic freq response and the prettiest blue glow you have ever seen. They have the same specs as the KT66 as I understand

Thank you very much for those interesting insights, I look forward to trying these

I think your instincts are correct. If you want to make a jump in sound quality… your integrated amp is the place to look.

First question… integrated or separates. If you want to keep moving over time… then you want to go to a separate preamp and amp. Given what you have I would think upgrading to Audio Research would be the obvious next step. Either integrated or separates… Separates would be best. Alternatively Conrad Johnson… for a slightly warmer presentation… or VaC.

I also would say the Prima Luna. I, unlike others, dont like pentodes in power amps. The Hana is a good cartridge but not great. I wish I knew more about your phono preamp. Dont forget about your cables.