WARNING: The following contains spoilers for Season 1, Part 4 Episode 4, Episode 9 and Episode 10 of Pokémon Journeys: The Series, "A Crackling Raid Battle!", "Sword and Shield... The Legends Awaken!", and "Getting More Than You Battled For!" now streaming on Netflix.
In Part 4 of Pokémon Journeys, Ash and Goh are encountering more Legendary Pokémon than ever before. Goh's goal is to catch 'em all so, inevitably, he is going to encounter many more extremely powerful and rare Legendary Pokémon.
However, these many of these Pokémon are important to the ecosystems and wellbeing of the Pokémon universe. Legendaries typically join the team of a trainer they deem worthy but, even so, is it right for him to catch them?
In Part 4 Episode 4 of Pokémon Journeys, Ash and Goh raid battle with Team Rocket against a Zapdos. This is the first instance we get of Goh actively trying to catch a Legendary Pokémon, however, Zapdos breaks free and flies off before he can catch it. Many Legendary Pokémon are related to weather phenomena, including Zapdos. During Part 4 Episode 4, Zapdos was resting at the Power Plant after having produced thunderstorms over Vermilion Harbor.
It's unclear if Goh will ever encounter Zapdos again, but if he does, it seems likely he would catch it solely for its Pokédex entry and then immediately release it. Though Zapdos isn't as important to the environment as, say, Groudon or Kyogre, producing thunderstorms and electricity still serves a purpose, and Zapdos may be needed to make such phenomena fully functional.
Part 4 of Pokémon Journeys also involves a mini-arc going through the Sword & Shield games' plot. In the final episode of the arc, Ash and Goh face off against the Legendary Pokémon Eternatus. Like in the games, the trainers need to borrow the power of the Legendary Pokémon Zacian and Zamazenta to stop it. However, the boys don't just defeat the Pokémon -- Goh and Ash throw a giant Pokéball out and manage to catch Eternatus, registering it under Goh's Pokédex.
Unfortunately for Eternatus, it needs to be eternally asleep for the Galar region to function. Eternatus is the source of the energy needed to Dynamax and Gigantamax, and when it's awake its powers leak out uncontrollably, causing Pokémon across the region to randomly Dynamax. Thus, Goh can not reasonably use this Pokémon in battle. At the same time, releasing Eternatus to the wild would cause mass mayhem again. So, the only solution is to keep it locked up under Professor Magnolia's care. Being trapped in a Pokéball for the rest of eternity is a rather sad existence though, so perhaps in the future Ash and Goh can figure out a better way for Eternatus to live on.
The last Legendary Pokémon Ash and Goh encountered in Part 4 was none other than Mewtwo. Naturally, Mewtwo is far too powerful for Ash and Goh, but the two trainers managed to hold their own in battle against the Pokémon. At first glance, it might seem like Mewtwo is a Legendary up for grabs. What purpose does Mewtwo serve now that it is no longer under Team Rocket and Giovanni's grasp?
But, when Ash and Goh arrive on Mewtwo's island, they discover a shy gaggle of Pokémon, including Mareep, Buneary, Skitty, Skiddo, Mudbray and Patrat. This may seem like a random ensemble of Pokémon, but they all reference animals regularly used for cloning research. Mewtwo has realized its reason for existing is to assist Pokémon like these that have been poorly treated by humans and create an environment for them that is human-free. Once Ash and Goh realize this and are satisfied with their battle, they head off. With Mewtwo being related to Mew, it's possible the boys may accidentally encounter Mewtwo again on their journey, however, with the amount of responsibility Mewtwo now has, it seems unlikely Goh will want to keep the Pokémon.
Trainers who own Legendary and Mythical Pokémon aren't a complete anomaly. Ash had many infamous encounters during the Diamond and Pearl series with trainers like Tobias, who owned a Darkrai and Latios, and a Nurse Joy who owned a Latias. In an earlier episode of Pokémon Journeys two Celebi, a regular and Shiny one referencing the Koko movie, briefly appeared, showing that, at least for some of Legendaries, more than one of them exist in-universe.
It's entirely possible Goh may catch and keep a Legendary or Mythical Pokémon eventually, especially if he realizes there are enough of them that he isn't terribly affecting any sort of balance. Most Legendary Pokémon don't belong in the hands of a child, however much like Ash, Goh is good-hearted and can be trusted to either release a being that is too powerful for himself, or entrust it to someone else, much like he has done with Eternatus.