I'm also somewhat skeptical that every new web app needs to be an SPA but one thing I'm 100% sold on as a generalist with the bulk of his experience on the client side is that a service-oriented architecture that hands off raw data rather than HTML to the client is the way to go whether you're loading prebuilt pages/views from the server and doing a lot of dynamic stuff with data after page load or building almost everything 100% with JavaScript.
The reasons this is preferable to a client-side dev are much the same as the reasons nobody wants HTML in the database. What's the client-side dev supposed to do when they want to resort a table for instance when they've been handed HTML? The performance cost of handling all that on the client is trivial compared to making another server request to do that for you. Also, HTML-building is pretty well-covered in JS-land. Sorting data and building new HTML table rows out of it is pretty trivial work for an experienced client-side dev.
And of what use is the back end architecture to a front end for another device that might need to do something exotic like implement widgets that are 100% canvas or a totally different HTML structure? Why should a client-side dev have to load up visual studio or knock on the back end devs door to make a strictly presentational tweak?
As for your concerns about the loss of strongly typed template validation, trust me when I say that if you're dealing with a competent client-side dev, you will find no .NET framework or visual studio tool that is more coal-to-diamond-to-dust-crushingly anal about well-formed, valid HTML than s/he is.
From the full-stack perspective, I like it because it means I'll never have to fish for business or app logic some yutz decided to drop into the templating layer. Not to mention the per-user load it takes off of your servers while in many cases actually improving load experience for the user in modern browsers with modern computers.
I think it's also easier to reason about the back end architecture when you've fully isolated it from all the presentation stuff. You're no longer grabbing data to mash it into some HTML. You're pulling it together to create an implementation-independent data structure that concerns itself more with general use than what's going to be done with it on the other side. IMO, that tends to lead to more consistency in how things are handled since the data is now an end goal rather than the second to last step of a process and there's fewer opportunities for unrelated concerns to get their wires crossed.
I keep view models in the web project, for reason you stated, it's usually only useful to the relevant view.
I'm not sure why your data access layer would reference the web project though?
Best Answer
It's tempting to ruthlessly eliminate duplication in our systems, and for very good reasons, but we must always ask ourselves whether this is real duplication or apparent duplication.
What I mean is that we'll sometimes have code that looks identical, but each will have their own reason to change, independent of each other. If you think the view model for the App will always change in tandem with the view model for the API, then they should share the same view models. However, you'll likely soon find that the view model for your app needs a property that the API doesn't need, due to needing some ephemeral UI element (like, say... a drop down list that contains data from some external system you'll be "borrowing" in your app).
I see the app and the API as two different front ends to the same business models. They should absolutely share the same business models (logic/dtos), but sharing view models is iffy at best IMO. If things are identical now, you could share the code, then inherit from your shared classes later, if you run across a situation where you need to extend the base models. Maybe the best of both worlds.