What? 'Generation' is a term that even The Pokémon Company uses in official statements, so I don't get where you got the idea that it's completely unofficial. Also, I'm surprised you think it's 'outdated' and has 'started to lose value' when it still does its job as a basic categorization quite adequately.
It's unofficial. That TPC very occasionally use it doesn't make it any less fan-created, it just means they've used it a couple times because their audience uses it, because it's part of the language that players have made. See the Bulbapedia page on the term for example - in my opinion, two press releases, a tweet, and a blog post is very sparing usage for going on three decades of business. It's not really like "Shiny," for example, which since around Gen 4
has become widely used official terminology with a specific and clear meaning.
I say it's outdated and has started to lose value, in my opinion, because it's a system that was created when the series was more rigid and stuck to more hard-and-fast rules, and it has limitations that are rooted in that origin. Maybe a more positive way to phrase it is that the concept of a generation has evolved past what I think it originally was.
Take the seventh generation, for example - that's when I think things started to go a little weird. In terms of just the main series, it's comprised of two sets of games on the same system running the same engine - which is perfectly normal for a "generation" - and then also a set of games that is on a new engine with a new graphical style and which doesn't include (almost) any Pokemon outside the original 151, all on a completely new console generation. And as a note, both USUM and LGPE introduced new species. In older generations, jumping consoles was reserved for spin-offs instead of main series entries, and introducing new species didn't happen until a new generation came. One could even argue that the edges started to fray a little as early as ORAS introducing new Mega Evolutions - what's allowed to happen within a generation has seen a gradual change.
The term "generation" has thus lost utility in my eyes: where it used to encapsulate a rather cohesive set of parameters, like visual style and console and National Dex list, it no longer directly implies any of those things. It's just a marker in time: some point at which a pair of games is released that's in a new region (but not a totally new version and time period an old region, for whatever reason) that also introduces new species and mechanics (but those parts can happen intra-generationally too). Anecdotally, I've experienced more difficulty in using the term "generation" lately with my few IRL friends who are deeply invested in the series; I'll say Gen 8 and they'll think I mean just Sword and Shield, or I'll say Gen 7 and they won't know that it includes LGPE, when these are people who would have known exactly which games I meant back in 2017 or so when I said Gen 7 or 5 or 2.
I don't expect anyone to agree, and I realize it's a little nitpicky and doesn't ultimately mean much. I also realize that the main games that throw a wrench in things - LGPE and Legends - might be unique one-off oddities that don't mean much long-term. It's all just my observation as someone who's been a fan for over two decades about how I've personally experienced the evolving context of the term.