I agree, the nu-Who gets way too much stick for being about romances.
The Rose one grew over the course of their time together. She originally really didn’t like this new guy who’d replaced her Doctor, but they spent time together, they had fun together and he started to feel something he probably hadn’t felt in a couple of hundred years at least. It also ties into Tennant being the most human of the Doctor’s incarnations. I wouldn’t say it’s what every episode in their season is about though, it’s more just a thing that happens around the plots.
I don’t think the show ever tried to replicate that. Martha season just isn’t very good (although she’s okay) but the point that the Catherine Tate season is directly setup to show they are just mates palling around the universe together.
And while there is a hint of something with Amy and Matt Smith, when you watch those episodes back-to-back (as I have recently done) you realise it’s over quite quickly really and then it’s more about Rory’s insecurities. But Amy chooses Rory pretty explicitly early on and then its very much the three of them together. The Doctor loves them both, but not in a romantic way.
River and Matt Smith is an entirely different thing and it’s brilliant, but yeah, definitely a romance but not the focus of every episode at all.
There’s a bit of jokey romance between Clara #2 (Victorian governess) and the Doctor but I don’t think there is with #3 (actual 2016-era Clara).
I guess you could say there’s a romantic vibe between Capaldi and Missy, but played very much as these are two Elders of the universe who are just a bit weary and glad to have someone like them around.
Edit: Interestingly, I’d previously always thought that Tennant was my favourite Doctor, but doing the rewatch with my wife recently it became clear that actually it’s Smith. There’s just so much bad filler in Tennant’s run that when we got to Smith and the Ponds it was like we both just sighed with relief. Smith is just so much fun in the role and, personally, I always loved the continuing plot - although I admit it works a lot better when you watch them all close together rather than having to wait months, if not years, between sections.