Php – AJAX or web sockets for client-server communication

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.

As is often the case, it depends

With a pure JS framework

• you push a lot of processing to the clients. That means your webserver for the client application will only be serving static files that can be cached.
• your clients have to be able to handle that processing and need some amount of computing power
• your clients need to fully support all JS-features that your front-end framework uses and you are somewhat tied to the support that framework offers for older browsers. If your framework's newest version decides to drop support for some older IE version that your clients still rely on, you will be stuck with that.
• your REST API's endpoints needs to be accessible by all the clients which might be an issue

Having an intermediate layer means

• you put the processing power in your single server, which is something you can control and, if need be, add more resources to. For clients this is more difficult and sometimes impossible
• you have the ability to control the calls to the REST API and, for example, cache data so you don't need to hit the API every time
• you have more control over security as the endpoints for the REST API do not need to be public
• you have more fine-grained control over the rendered HTML and are not tied to the components provided by your framework. You could create a component that initially uses JS to build part of a view and when performance becomes an issue replace that by HTML generated from the server
• you need to run and maintain a server with some server-side programming language capability

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.