Catching pokemon in a pen and paper game

Related to this and this. If I am using tamed monsters as a magic system, there has to be a way to catch them. Or not? I am very confused about how to do it. 

On one hand I have the analogic ways:

Getting into the wilderness (aka the tall grass)  until a monster pops up, then catching them with a pokeball equivalent. This works perfectly on the videogames, but I see how it can become tedious on a tabletop. Grinding a place to find one pokemon that one player happens to want makes all the other players who dont have anything to do at that place just stop and watch the show. If I do it too easy so the pokemon is catched fast, its almost the same as giving the player the monster straight. 

An even more analogic version of this is if the monsters are not catch with pokeballs, but convinced to follow a leader by actually roleplaying and discovering the monster's motivations or personality. This is great to have sometimes, but having it as the main point of the game for sorcerers and not for the rest of the classes might make them absorb a lot of game time that is not spent doing teamwork. It is also difficult, as it might seem easy to bond with a pikachu in game, but other monsters can be a challenge for the GM to roleplay in this manner.

A middle option is to make the player roll their pokeball or do their talk no jutsu once per downtime or so, allowing retrys once some time has passed.

The more boring option is just to give the players pokemon on level ups just as if they were magician spells. They can select from a random heap of them (whatever invocation scrolls are available on the local library, or whatever local pokemon are on the nearby grass), maybe adding any monster they met on their adventures to the list of possibles. This removes the need of doing all the work in character, though still the first options can be used for legendary or special pokemon. But something feels bad about taking from players the experience of catching their rattattas.

This opens the problem of that players catching their pokemon diegetically advance more than those spending their level ups on them. Its easy to fix, though, by making those diegetical ones "temporal" until the next level up, and then requiring the player to spend the level up "keeping" them just as if they were their downtime catch. This can apply from cosmic monsters who manifest as spells to a plain dog who is an animal companion. 

As an addendum, there is something I don't like about Naruto's summoning: Servant animals are supposed to be someplace when you just "teleport" them into battle or whatever. This is ridiculously easy to break by PCs in order, for example, to send real-time messages through them as they dispell the summoning; and you run into the risk of summoning a sleeping dog at night or summon a giant toad at the moment he was sitting on the toilet taking a shit,