Yeah and slow down every time connection is a bit slower or you want to search something across whole project. Very often I am doing search trough all files directly in Sublime and I can do that because I am restricted only by throughput of my SSD and CPU.
If you're searching across a whole project, the server is footing the brunt of that. it's only sending back the results. That's not usually a lot of data, so your connection would have to be in the kilobits to be that big of a burden.
This is simply not true. If you search using external application, all the work is done locally, not server site. During the search Sublime have to open EVERY file, search for word and then close it. That's a lot of data going back and forward. Of course search can be done using server tools like grep, then the search will look just as you said, server will do all the work and then return result. There are two problems with that approach, first this is not as nice user experience as it is in Sublime, second: server (usually VPS) is in most cases slower than my dev machine, because not everybody works with multi cpu behemoths.