What they care about:
- Does it do what I need it to
- Can I access it easily
- Is it easy to navigate
- Does it work or is it buggy
- Is it slow
- Does it abuse notifications
- Is it prompting me to log in all the time
- Are there too many ads
- Can I use it on any device I have
- Does it complain about updating constantly
I don't think any of those actually has to do with being a website. The problem is that web browsers and websites didn't get a not-stupid UI for phones faster than the phone vendors could funnel everyone into their walled garden. Remember "click here for mobile" and having to develop two web interfaces? And half the time functions would be missing on one side?