In general I agree, if I have to choose one I prefer lockdown defense, but the true-blue unicorn in my opinion is the rare player who can do both at an elite level
It's very rare, I'm not the kind of person who thinks there's 32 #1D/#1C ect because there's that many teams, in my mind that distinction belongs to players who are the ideal version of fitting that role
Probably not a comprehensive list but I'll take a shot at it in no particular order, just sorted by even strength ice time the last couple years and combed through the names
Heiskanen
Dobson (short track record but he's tracking for it)
McAvoy
Hedman (declining now, was basically the prototype for a decade)
Fox
Makar
That might be it
There are a lot of great defenseman who provide either offense or defense at an elite level but are lacking on the other side of the puck like a Josi or a Slavin, who are excellent players, but they aren't truly the best option in all situations
There are also very well rounded D who are really great at both but are a touch below the elite of the league like Lindholm and Morrisey, I wouldn't call them #1s either
I'm pretty exclusive with this designation, it's a short list
Noah Dobson is extremely mediocre defensively, McAvoy is prone to brain farts in critical moments, and Hedman has always been overrated defensively. Prime Burns was better defensively than all those guys. Arguably better than Fox and Makar too.
By your definition, I think Miro Heiskanen is the only real #1D in the league. So my next question would be, if the argument is that you don't need a true #1D to win a Stanley Cup, why is your definition so arbitrary?
I'm also not a "32 #1 defensemen exist by definition" person, but I judge based on overall impact more than being elite in every facet. I think that every great team needs a defenseman who can eat tons of minutes in all situations comfortably. I think every great team needs a defenseman who is dynamic offensively and can put up 60+ points regularly. And I think that every great team needs a defenseman who can lock down leads, win tough matchups, and generally shut down opposing star forwards. The catch is that I don't think all those roles need to be (or even should!) be packaged into one player. In fact, I would argue that the only NHL defensemen who can do all those things is Miro Heiskanen, but he isn't top-10 at any of them. I would personally argue for Josh Morrissey's inclusion if you're including Heiskanen.
My argument for defining a "true #1D" is that they fulfill two of those archetypes (typically the offense or defense plus minute-munching) and the archetype that they don't fulfill is drowned out by them being elite at the one that they do.
Burns could eat 30 minutes comfortably and was elite in a singular way offensively. Sure, he had his defensive shortcomings, but his impact on the ice was overwhelmingly positive and he could be counted on in all situations even if he wasn't elite defensively. Karlsson in his prime was the same way. I can buy arguments that Sharks Karlsson wasn't a true #1D but Senators Karlsson absolutely was.
I don't care if you win the game 6-3, 4-2, or 2-1. Just win the game.