The Classic servers aren't P2W because they're subscription-based.
But from my experience on Grand Crusade on Innova, it was the most hardcore P2W I've ever experienced for the 10+ years I've been playing Lineage 2. It's really taking the whole thing to a next level - it's not as much pay to win as it's pay to play, because unless you buy ton of jewel boxes from the cash shop or buy adena directly from botters, you make basically zero progress. It's obviously P2W all the way through highest levels as on this chronicle there's so much room for improvement you can spend 2,000$ and still not be 100%. It's P2W in it's clearest form - buy jewel boxes from ingame shop for 100$, sell them to players (everyone's buying 24/7) and buy top equip from auction.
But anyway the point is, as you mentioned, official servers give you two chronicles - Classic and Grand Crusade. One is subscription-based extreme hardcore chronicle where you farm all day to make few % and the other one is barely Lineage 2 with insane P2W. High Five is standing nicely between these two (still a solid Lineage 2 vibe in it while tons of stuff to do), but offic simply doesn't give us option to play it.
Also I don't think you can generalize private servers like this - sure when there's some homemade L2j serbidor opened by 10 year old it's probably not going to be 100% experience but if you know where to look at, there're a few high quality servers - RPG Club I consider one (even just for regular server merges), Classic Club is up for 2 year too, Dawn (unfortunately RIP now but was a good run too), Warpgate (the guy really seem to care but not sure how far he's with the developement) or PlayInera (these guys really seem to work hard)...
Plenty I didn't mention but my point is that even after trying Grand Crusade on Innova I still feel like some private servers are much better. But that's just my 2 cents.