Pokemon would probably be a disastrous flop if it was introduced in modern times if they introduced it in the same way.
Sword and Shield (which I will assume would be what would be released, if we're going with this theory) are phenomenally bad quality titles that are held up solely because they bear the Pokemon trademark, and their flaws would be highlighted and lambasted if they were not a part of a very old and very popular franchise that has recycled the exact same elements for its entire lifetime. For better or worse, the AAA industry - because Pokemon IS a AAA title, like it or not, and any and every Nintendo-associated IP gets put in a very prominent spotlight - focuses on things that Pokemon does not have, and has never had. High quality visuals, polished gameplay mechanics, a good narrative; Pokemon has none of these things. But it's what people expect by now, so Game Freak can get away with shit no new franchise just starting out would be able to in this day and age.
The assumption that there would be no monster-catching games without Pokemon can't even be made in this scenario because the concept was introduced by Dragon Quest V, which was released in 1992. Let's be clear here: the monster-catching genre did NOT originate with Pokemon. It was popularised by it, to be sure, but it's not an idea that would not be around if not for Pokemon. With the popularity of gacha mobile titles that have similar collect-em-all mechanics, it's not a huge leap to assume that someone else wouldn't have filled the void left in Pokemon's absence. Digimon would definitely still be around in its current format because it started life as a competitor to Tamagotchi, and in reality the Digimon games are absolutely nothing like Pokemon; the similarities between the two franchises are surface-level only. But the shoe would probably be on the other foot with the negative comparisons despite that, perhaps even more so, because Digimon titles are very narrative-driven and offer a great deal more flexibility and complexity with their gameplay systems. Although Digimon has done very well in carving out its own niche and growing up with its fans - something Pokemon definitely has not done - so there's nothing to say Pokemon couldn't do the same...likely it would fill a similar role that Yokai Watch does now. If they marketed Pokemon solely towards kids, rather than trying to cater to both old and new fans (after all, they wouldn't HAVE any old fans to cater to!) then the possibility that Pokemon could enjoy Yokai Watch's level of success is fair.
But Pokemon would likely be viewed as a better value, but significantly lower budget and lower effort, version of a gacha mobile title, as it has the same sort of collect-a-thon nature. So Pokemon Go would probably do well if it was introduced in modern times (although perhaps not quite as well as it has) but if they released a game like Sword and Shield as the first one, the franchise would fail in the biggest way possible. SnS are not console-quality titles, and if they're going into a market where similar games exist...well. There's no way they would be even remotely successful. If they released Let's Go Pikachu and Eevee after Pokemon Go I think they would do reasonably well, though - the games are well-adapted in terms of Go's mechanics, and they are of a much higher quality than Sword and Shield generally. They could build off of that success, although again, they're still notably lacking and would never attain the kind of popularity Pokemon enjoys now.
If they started with the first generation now...well, if they were released as they were on Gameboy, they'd receive very heavy criticism too because my GOD the originals are dated as hell now, but given the popularity of indie titles, I think classic Pokemon titles would fare reasonably well if they were introduced as indie games. I doubt they'd gain the success of, say, Stardew Valley, but I think they'd do well as niche games if expectations were modest. The lack of an online component would severely hurt them though.
I've only talked about the games here thus far, but the games are about 20% of the franchise's total sales. Plenty of very successful anime series have trashy video games made from them (seriously, there are a tiny handful of good anime-based video games) so even if the games were shown as the trash they are, there's nothing to say that the anime still wouldn't be hugely successful, which would mean that the merchandise would be hugely successful, which would mean the games would sell far more than they have any right to irrespective of the quality of them. If Pokemon were being introduced now, as long as it didn't start life as a video game series, it would probably be able to do significantly well. Probably not largest franchise in existence well, but I'd expect it to do better than, say, the MCU, which is 10th on the list of highest-grossing media franchises. In fact, if we take the video game sales out of it, it still comes in at the third-highest. The merchandise is worth $64 billion by itself! So yeah, if they went through literally any other avenue that isn't video games, Pokemon could do almost as well as it does now in this scenario. But if we're just saying "What if Pokemon games came out now" then no, they'd fail horribly.