Alright, after some more tries I finally managed to get it working by following this [Support Guide] DNS Quick Start - How to set up DNS sent by @amelia and some additional bits.
TLDR: I stopped using Netlify DNS, and instead followed the External DNS guide in the Support guide, and I delegated it to studio4web.com where I get my email from and then added the records there.
(I may have misunderstood somethings along the way, so correct me if I’m wrong…)
Basically I’ve been using 3 services. Netlify for deploying, Namecheap where I bought my domain and then studio4web where I have my CPanel, email host etc.
What I needed to do to get Netlify DNS working was to add the nameservers dn1-4 to NameCheap, and add the necessary records for email etc. to Netlify. And that all worked, however the studio4web host also requires me to add their nameservers to Namecheap to get it working properly (because I also had to register my domain through them to get emails).
However, after doing all that the mail wasn’t being sent, like I stated above, neither when I had the studio4web nameservers added nor when I didn’t (@coelmay). Although even that needed some additional configuration with studio4web because they were providing some kind of SSL which then caused an INVALID_CERT_ERROR when accessing the page.
The second thing I tried was using NameCheap DNS, which again worked for the website, but not for the email. Because, like I said, I need to add studio4web servers to namecheap, but when using their Basic DNS this options is blocked. And even adding all the needed records didn’t help.
Lastly I decided to use studio4web DNS. Which means I added the name servers to NameCheap, and then added A and CNAME records there. And mail records were not needed since they provide it as well. Now everything works.
Sadly I didn’t get much information about why I had all this trouble. It could even be that I didn’t wait long enough for everything to propagate before trying… Maybe some of you who are better at this might see something in my description… But anyways, all works now, thanks for all the help