In theory, this is true. In practice, it's just hit or miss. I think it's just hard to maintain web across all the different platform and screen sizes. It's probably due to the fact that an iOS developer specializes in one platform and most web development shops will have to support all platforms.
"If Apple would allow web notifications"
I do not use apps because of notifications. I use apps because in general, the performance and experience is much better.
For example, you can create "app" that is just a link to browser. There are plenty of action you don't want random web-apps to perform due to security/sandboxing issues, but if you can grant explicit permissions that issue goes away. It is one of the reasons think like election exist. But could apple/goog provide a better electron that was built into the OS?
Even apps without notifications are still apps, because they can take additional data from users, sometimes even location and the whole address book.
Look at sites like reddit for exmaple, which purpusefully cripple the mobile web experience to get the user to install the app.
The problem is that purchases would no longer be subject to Apple's 10x industry standard (30% instead of 3%) captive credit card payment processing service (for apps), so no web notifications in MobileSafari that might undermine platform control and payment processing monopoly. Apple, of course, will say this is about protecting users from notification spam, and sidestep the inconvenient facts that it protects an Apple cash cow and locks developers and users into a closed and proprietary platform.