Looking at Pokemon sales per generation it seems like the series has had pretty strong growth on Switch compared to GBA/DS/3DS entries, almost reaching peak Pokemania sales on GB.

Pokemon Red/Blue-31.05 million

Pokemon Gold/Silver-23.73 million

Pokemon Ruby/Sapphire-16.22 million

Pokemon Diamond/Pearl-17.67 million

Pokemon Black/White-15.64 million

Pokemon X/Y-16.72 million

Pokemon Sun/Moon-16.32 million

Pokemon Sword/Shield-26.02 million

Pokemon Scarlet/Violet-23.23 million

The series had a steady decline from Gen 1-3, stability through Gen 7 and a revitalization in Gen 8-9. But this ignores the 3rd versions/sequels, adding them gives us.

Pokemon Red/Blue/Yellow-45.69 million

Pokemon Gold/Silver/Crystal-30.12 million

Pokemon Ruby/Sapphire/Emerald-23.28 million

Pokemon Diamond/Pearl/Platinum-25.27 million

Pokemon Black/White/BW2-24.16 million

Pokemon Sun/Moon/UltraS/M-25.51 million

Pokemon Sword/Shield-26.02 million

Pokemon Scarlet/Violet-23.23 million

Did Pokemon benefit from the Switch effect or did the shift to DLC expansions just eat up the 3rd version/sequel sales?

Though I have issues with the Pokemon games on Switch, I believe there's no denying that the sales of the series have grown tremendously on Switch (as has nearly every Nintendo franchise that is on Switch compared to their predecessors).

I think DLC has helped actually, and your sales comparison at the end kind of proves that. SWSH managed to outsell Gens 3-7 even when including third versions (and SV likely will as well). On top of that, the Switch games cost more and we don't know how much of those 20m+ users purchased the DLC. Meanwhile with third versions, not everyone would double dip. Like if you had Diamond or Pearl, you might have thought Platinum was pointless (or you'd skip DP and wait for Pt).

In the Switch era, you'd get SWSH or SV and then keep all your data to be able to play more content through DLC.

No... Pokemon is a shadow of Its former self.

It did, got a ton of games too.

Sales-wise its very solid.
Game-wise tho... eh.

"Did Pokemon benefit from the Switch effect or did the shift to DLC expansions just eat up the 3rd version/sequel sales?"

Ummmm... that's a really weird question, because those two options don't seem to in any way contradict each other. Pokemon seems clearly to be selling more. I don't see why the fact that a chunk of the sales are DLC should impact things.

I think it's hard to argue Pokemon is selling better. In addition to the games listed, you have Pokemon Pearl/Diamond, Lets GO, and Arceus selling around 15 million each. If you look at sales per sku, or per year, I'm pretty sure they're selling more games even before accounting for DLC, and when you throw that in they're selling waaaaaay more content. The fact that they're not using the same exact model as they did in the past shouldn't really matter.

To an extent but it's growth has been minimal compared to most other series. Before Switch is was probably their second biggest seller behind Mario Kart, now it's behind Animal Crossing, Smash, Zelda and more or less tied with 3D Mario.

Yeah, it definitely didn't get the same Switch effect as to the other franchises in sales
If they actually step up their game they would be reaching new heights

Speaking for myself I was not interested in Pokemon for a long time. In fact, I was not interested since they expanded the initial ~150 rosters and kept reusing much of the same formula. However, I see potential in the new approach even though the execution of the first title is a tad to be desired, it is still the right direction in my opinion and I'm definitively interested.
I do hope that eventually, they'll do a remake of the original with the new engine when they master it. Just like they did with FireRed & LeafGreen.

When counting the third versions, you'll have to consider that a non-insignificant amount of sales will be to players who are double, triple or even quadruple dipping, so even if total sales per generation remain similar, the actual playerbase is likely smaller for generations with third versions. Additionally, we can look at the sales of the remakes, which show a clear upward trend.

Firered/Leafgreen: 12.00 M
Heartgold/Soulsilver: 12.72 M
Omega Ruby/Alpha Sapphire: 14.60 M
Let's Go Pikachu/Eevee: 15.07 M
Brilliant Diamond/Shining Pearl: 15.06 M

Bonus: Legends Arceus, which is considered part of the main series, has sold 14.83 M copies. This is the most any single Pokémon game has sold, and since it's completely stand-alone, there are no double-dipping shenanigans going on here.