Do you care the method you catch a pokemon?

Nope.

I do to some degree, and it stems from my collecting rare ball 'mons in the main series (things like dream ball mons, apricorn balls, ect). The problem is that we don't have any control over eggs or raids, and 90% of the time I have no idea if a 'mon that I catch in the wild will be worth raising, so it's hard to really have control over it.

My catching method:

Pokeballs for Go Plus
Great balls for normal manual catches
Ultra balls for catches that I care about

This is something I care more about in the main series than I do in Go. With the amount of available balls being so limited and the fact that any given Pokemon can just flee if the catch isn't successful means that I'm more concerned with just catching it than about what color its home is. When you consider you don't have a say in what ball raid-exclusive Pokemon are caught in it doesn't make it worthwhile to fret about.

Amusingly, I do make a point to try and catch main series legendaries exclusively in Premier Balls. Tapu Fini with its Aqua Ring can go straight to hell for being a pain in the ass. Same for Kyogre in HeartGold.

My friends son catches all his shiny legendaries with his nose

Pokeball absol would be top tier for me