Thing is...there isn't any singular "Beatles guitar" sound that they always had on everything.
Maybe if you break things down into the various periods they went through as their styles evolved...but even there they had some variation, and IMO, it's not something that really requires a huge amount of effort and specific vintage gear or specialty boxes.
It's just tones...and once you identify them, a lot of gear can be dialed in to get you there.
I think WAY too much is made of the need to use the same "vintage" gear (or same any gear) in order to mimic a sound.
Also...just how identical is it or does it really need to be?
IMO... it's always going to be more about capturing the essence rather than dialing in the exact same thing with the same gear...and much of that essence is more about other things than just the gear.
Granted, having the exact same gear might make it easier/quicker...but again, I think it's just about identifying the tones and then dialing them in, and there's probably a lot of ways to get there.
And of course...as you found out...it's a lot to do with the players.
They tend to always sound like themselves, no matter what gear they use.