Hangouts does, but it's impossible to know if we're "supposed" to be using Hangouts. A year or so ago I received a pop-up in my Hangouts app that Google recommended I check out Messenger [0]. It also seems to be the currently recommended option [1] although I don't know why since, as you say, Hangouts can handle all those scenarios.
This has understandably led to no end of confusion among Fi/Voice users, including myself. [2][3][4][5]
Hangouts can't receive MMS without special support from your mobile operator. Operators such as T-Mobile and... Google Voice... have been known to insufficiently provide that support.
Okay, you can announce that everything is fine, but it just ain't so. I missed MMS messages on T-Mobile in 2016.
Google's suggested solution, besides twiddling cellular network settings and asking your friends to keep texting you until it works, is to stop using Hangouts [1].
I believe you, but it's something that isn't quite as easy to compare anymore. I have a friend with an iPhone on verizon that has missed MMS or found it delayed by days. GV is easier to blame usually since it's different than other services.
Yes, features have been removed that don't fit the repositioning of Hangouts as an enterprise messaging app, and bigger changes based on that repositioning have been signalled as coming soon.
Hangouts does not ship with the Pixel, you have to download it separately. IIRC it was also not easy to find in the Play Store.
The Pixel pushes you to use Allo which is based on phone numbers and your phone's contact list, not your Google account contacts. If I want to message someone on Hangouts but I only know their email address (which pops up helpfully if I have corresponded with this person in Gmail), no problem. You can't do that in Allo - the people you message are identified by their phone number, not the fact that they have a Google account. In fact, you can't use Allo until you register your own number. So what is Allo? Is it SMS? Hangouts could already handle SMS and Google also released Messenger intending it to be your phone's default SMS client. The only entity you can message in Allo that doesn't need a phone number is Google Assistant, which isn't that great. If Allo requires you to be identified by a phone number, how is it better than SMS? If I change my phone number, am I just hosed? The best thing about Hangouts is that it let you message damn near anyone in any manner while being platform independent.
Not only are they tied to a single phone number, but my understanding is you can't run the same app on multiple devices. And that's my ENTIRE google voice use case! I want a number that works wherever I am, whatever carrier I use.
Hangouts on my Nexus 6 has the same options.
Imgur link to screenshot -> http://imgur.com/a/vTlqO