Using Brave right now and no stupid popups. Make sure you turn off "Brave ads" or whatever causes popups. Always scan through every option in Settings at least once and turn all the junk off. The only mild annoyance left after doing this is that sometimes the new tab page has a little box shilling some Crypto coin biz which you can then silence. If this is all I have to suffer in order to fund a solid de-googled Chromium it's not too bad.
We don't put any popups anywhere by default. Are you sure the website wasn't trying, which flies a permission prompt same as in Chrome and other browsers?
I turned on the special, approved Brave ads thing last time I used it because I wanted to support Brave; and they only showed up as little pop ups on the systray and were infinitely more annoying to me than a static banner ad.
You click on them to open new tab pages if interested, can thumbs down and lower frequency in ads settings. You also get 70% of the sponsored images shown in 1 of 4 new tab pages. Static banner isn’t going to perform as well, so I hope you will give ads a try again. You could even turn off notifications but keep the sponsored images.
I think those ads are much better. In-page ads interrupt my content, claim my attention when I want to see something else, and distract from the page (and sometimes even break it). Ad-heavy pages are nearly unusable, especially on lower-power mobile devices. OTOH, I don't mind a popup once per hour - it's not modal, does not require me to do anything, does not consume resources and I can easily deal with it whenever I want. Page popup ads are infuriating, but Brave ones never really bothered me.