You can have a separate icon for that as though it were an app, and if you really want to, you can enable push notifications just as you could with an app.
Given that others have reported it, and it's been there for a long time, I suspect Twitter at the very least intentionally doesn't put too much resources behind the web site to force people to use the app.
Of course, if a service really wants to push an app onto me, it's clear that the app gives them some real benefits, and its usually the kind that aren't a benefit for me (more tracking, better ways to push ads, more "engagement" notifications, ...). So the harder something wants to push an app, the clearer it is that I never, ever want their app to touch my device.
I thought I was the only person dealing with this.. It happens for pretty much every tweet that I open in a browser. After a refresh, everything loads fine, but it's quite annoying.
In many countries, it is not even out of warranty!
Yes, some people change phones every year, but that does not mean the device itself is old just because you change it.
Although now that I think of it, maybe that's not such a good idea if you're using Android, given the lack of moderation on Google Play. I don't know what the third party Twitter app ecosystem is like on there, but I'd be cautious and meticulously vet the developer of any app you're considering trying.