caveat first so as not to get talked down to/waste time: weather is of course variable day to day, particularly so in the Netherlands.
Now then OP, in my estimate it's a tough call. I live in a similar wet, cool-cold winter climate to the Netherlands. Not as many, but plenty of day when it is both wet/rainy and in the 30's.
Some people are very into dry feet. Even in light precipitation and warmer weather they recommend waterproof shoes. I'm on the other end of the spectrum: like Pacific Crest Trail through hikers et al, in my experience it's better to have non-waterproof shoes that dry quickly, and quick drying sock. It rained (and rained, and rained, then rained some more) cycle touring all day outside in Europe last summer, and never once did feel like I'd rather be in waterproof shoes.
The cold, however, makes it a harder call. 30's are no joke, think the inside of your refrigerator. Cold feet are a real potential.
On the other hand, I've backcountry backpacked 4 straight days in very cold rain and wet snow, same trail running shoes as always, zero memory of discomfort from cold feet. And we are talking dozens of hours in camp, not pumping blood hard the whole time.
And you can go to any Seattle high school on the coldest, wettest day of the year, and every kid, including my own, is in soaking wet Chuck Taylor's (or Airforce One's, etc) and cotton socks and they are fine. Thousands of them.
The weather might be variable enough that your trainers are fine. And then you are carrying uncomfortable shoes around for no reason.
So all considered, if it was me I'd take my usual travel trail runners, my quicker drying but still a bit warm socks (like a smart wool or balega synthetic etc). I'd very likely both have wet feet sometimes, and also be just fine. And if I did regret that choice, I'd buy some comfortable waterproof shoes in Amsterdam, which has a LOT of shoe stores. Yours from home are already uncomfortable, so no big loss rolling the dice on something new. Just go with a type that tends to offer immediate comfort (waterproof Crocs brand shoes for instance) if you decide you need waterproof.
Just be sure your shoes dry out relatively fast - sometimes a lot of softer foam in the upper can create sponge issues. And bring an extra pair of socks or two so you always have dry ones. Stay on top of the shoe and sock drying in your down times in your room. Remember to pull out insoles, know the "wring socks inside a towel" trick, and that the hair dryer is you friend on low settings, worse enemy on high (don't melt your shoes!).
There! If you end up needing foot amputations send me the bill :)