Buy European

I’d suggest, depending on weather and exposure, simply getting shoes that can get wet and easily dry out and keeping a pair for indoors your bag.

You're going to have a hard time finding fully-waterproof sneaker-style shoes because the nature of a sneaker implies holes for ventilation and a slit between the tongue and the upper.

I'm aware these aren't what you asked for, but since that doesn't exist you might have to consider them anyway:

You could get some cycling shoe covers, although they're intended to go over roadie cycling shoes for use with clipless pedals (3-post cleats), not regular sneakers.

You could also get mountain-biking shoes (2-post cleats) or trail-running shoes, but they won't be fully waterproof.

Personally, when I was commuting a decent distance by bike (far enough to justify using clipless pedals), I wore cycling sandals when it was warm enough and mountain-biking shoes when it wasn't, and then changed into dry non-athletic shoes that I kept at the office.

Mukishoes is out of portugal and they will also resole your shoes for you for cheap when the soles wear out.

https://mukishoes.com/product/raw-leather-brown/

They have a leather shoe that is kind of sneaker style, but oil it up with a leather conditioner and it is waterproof. I live in Belgium, so very rainy and my feet have yet to get wet in them using mink oil IIRC (Definitely buy the wool insole though because they have thin minimalist soles so it gets cold when temp dips <0)

Have you considered a shoe protection instead? Often rain pants cover most of the shoe anyways, so my shoes stay reasonably dry over 15-30 minute commutes. (Not sport shoes, those get wet on sight of rain, not even the need of an actual rain drop… -.- )

Yeah, have a look in decathlon. They have waterproof shoes which look like trainers. I've got myself some, I can go through puddles with them and stay dry.

I like my Ecco trainers that are waterproof, company is headquartered in Denmark. https://gb.ecco.com/

Not sure if it's meets your requirements, but I've had a pair of Salomon Speedcross shoes in various iterations over the years and only get a new pair when they finally stop being waterproof. Since they use Goretex (a problem ethically in its own way) it eventually breaks down and water starts seeping in, but it lasts as long as a running shoe normally does anyway

Vessi

There's a company here that makes and sells shoes out of recycled stuff, or so I'm told. They advertise a slip-on shoe that is waterproof, among a bunch of products.

Look for Vessi , either online or where you shop for shoes.

Wouldn't you want non-waterproof shoes that dry easily? Like canvas shoes? What shoes were you wearing to ruin them with rain

nothing like that exists.

buy waterproof shoe covers. that is what they are for.

I saw some nice leather sneakers last time I was shoe shopping. They were pretty expensive and I forgot the name of them.

I was thinking along the same lines of you and thought if you coupled it with a tub of Dubbin they would make great winter shoes.

In the end I settled with leather work boots and dubbin plus some nice, breathable runners for exercising.