As others have commented on your question, you need server-side functionality to provide effective and efficient search in a website; otherwise you would need to make all searchable content available to the client-side search engine; unless the volume of searchable content was very small, this would be troublesome.
Once you have the server-side functionality, yes, you can add AJAX to make searching niftier- esp. autocomplete functionality, quick filtering, etc., although that's also going to rely on the server-side functionality.
My personal point of view; if you do have the time to do AJAX and it adds speed, do so. But it is time-consuming and perhaps your time would be better spent some place else.
Quit thinking in terms of MVC. MVC alone only describes the interactions well if everything happens on the server-side. You need to think at a higher level of abstraction. For example, Data Layer (ex database), API layer (ex server-side), Client Layer (Webapp, IOS app, Android App).
A basic SPA works on the premise that the whole site runs on one big HTML file. Reloads don't require a round-trip to the server because you're basically showing/hiding different sections of code.
In practice that's not necessarily true because the framework (ex Angular) can pull template partials and data from URLs in the background. It sounds like you understand this much but can't figure out how concerns should be separated between server/client.
For templating, do it the same as you would in Django. Except, instead of loading the files from a local directory you fetch the partials from the server.
For generating views that include data you'll need to make a controller that can fetch the data and use it in the view partial. There are many ways to go about this but the most common is to simply create an API on the server-side that accepts data requests and responds with the results in JSON.
In MVC terms. The server and client will both still use models, views, and controllers:
On the server (ie the Data API). The models are the database/storage access classes. Instead of generating raw HTML the views will be used to format/serialize data for transfer. The controllers work out the messy details between the models and views.
On the client (ie the WebApp client). The models are the functions that handle fetching and deserializing data requests. The views are your html partials that get parsed into static HTML. The controllers /directives manage combining the two.
Can you still do HTML generation on the server-side?
Of course. How much work you decide to do in the client vs server depends on your goals.
Then what's the benefit of client-side frameworks?
If your goal is to create a SPA, then frameworks like Angular are a much more robust option than the alternitives which would consist of firing off a bunch of AJAX requests, then plugging it into the DOM using something like jQuery.
What's so special about client-side frameworks?
Nothing really, they push the responsibility of HTML generation into the client. Instead of splitting HTML templating and dynamic manipulation between the client and server you can now do both in the client. It provides a cleaner separation of concerns and reduces load on the server.
If you need a data layer that can be accesses by multiple platforms (ex IOS, website, Android) then you've likely isolated the data requests into their own API already. If you have that done already, pushing the client-side code to the client makes sense since you're already doing that for the other platforms.
Simplifying the server to just a data layer makes life a simpler for those who develop on multiple platforms. Client-side frameworks made the process of fetching data and generating HTML dynamically on the client much easier. If your goal is to build a SPA, using a framework will provide a good scaffold to build on.
The major downside of client-side HTML generation is that it relies on JavaScript and most search engine webcrawlers are too dumb to use JavaScript. That means, if you have a content-heavy site you're going to have a bad time trying to get search engines to index your content. There are hybrid approaches to conditionally load content generated from the server to webcrawlers but they add another additional layer of complexity so the tradeoffs likely outweigh the benefits.
Best Answer
As is often the case, it depends
With a pure JS framework
Having an intermediate layer means
The question is: which of these issues matter to you and which don't. If you have a requirement to support lower-end hardware I'd stay away from JS front-end frameworks. It's very hard to control performance that way and you are very limited in the optimisations you can perform.