Browsers don't allow notifications if you don't have the site open. Browser ads can get blocked by browser extensions. Browsers make it harder to have an icon for a site/service directly on the home screen. Browsers make it harder to get extensive permissions. Browsers allow content to displayed without first being run through an approval process.
For companies these are all downsides but for me they are all upsides. It really is us vs them when it comes to apps vs browsers. The only reason they offer websites at all is out of fear of losing a big chunk of users.
Browsers most definitely do allow for notifications if you don't have the site open. I use that feature all the time and it works perfectly.
Google Chrome does seem to catch spam sites that abuse notification permissions to send ads, though, so for a certain category of crapware websites aren't an option.
Darn as I wrote that I doubted myself.