Because he doesn't need to.
In the games, the in-universe reason for catching every type of Pokémon is to complete the Pokédex, to help the regional professor with their research (the out-of-universe reason is, of course, sheer completionism). The Pokédex does not contain complete information on a Pokémon species until you catch at least one Pokémon of that species. This is the same reason that Red, in Pokémon Origins, catches all 150 - because Oak asked him to, for research purposes.
In the anime, the Pokédex is already complete. Every single time Ash encounters a new Pokémon (with a few exceptions), the Pokédex lights up, shows its picture, and rattles off a short expository explanation. There's no reason for Ash to go around capturing one of every Pokémon, because neither he nor Oak needs one of every Pokémon.
Doesn't becoming a Pokémon Master mean catching them all?
No. The anime has never actually defined what you have to do to become a Pokémon Master, and likely never will, so that the writers can justify the series going on and on forever. It's worth noting that in the anime canon, many legendary Pokémon are one-of-a-kind, and some are so critical to their ecosystems that capturing them would have dire consequences. So "catching 'em all" is something only one person could ever do, and that person would devastate large areas in the process.
In summary, in neither the games nor the anime does capturing all 898 Pokémon (as of Gen VIII) make you a Pokémon Master. In the games, all you get is a lousy Diploma and a sense of completionism. In the anime, it's a completely impractical and pointless goal that, in spite of the series' tagline, isn't really worth pursuing at all.