If you need to do some more massive JavaScript/DOM manipulation and querying then calling getElementByID and the others repeatedly will lead to extremely long code. It will also lead to unmaintainable code if you just put everything in one big code block. To keep your code short and to enable easy reuse, you'll need to encapsulate this code into functions.
You do realize all of that applies when you use jQuery as well, right?
Maybe you don't...
. Once you start making functions whose purpose it is to manipulate the DOM in a similar way across many different browsers, you are better off going with jQuery.
In the context you're trying to support the use of jQuery, it very obviously offers you no benefits what-so-ever. But you know that, right?
Well, I can't blame you for trying to make such a non-argument. It's VERY difficult to defend the use of jQuery. See, what this all boils down to is that you're comfortable with jQuery, not so comfortable with JavaScript, and already know that jQuery has long outlived any utility that it may have had in the past. You don't want to see your favorite "tool" die, so you feel the need to protect it from reality.
jQuery has promised, but never safely delivered cross-browser support. With 2.0, it gave up on that goal entirely. Besides, once you drop support for IE6, there's very little you need to do to maintain cross-browser compatibility. So little, in fact, that using jQuery actually *increases* the amount of work you need to do as you get to deal with cross-browser problems in jQuery and cross-jQuery problems with your plugins!
Recently, I decided it was (long past) time to upgrade so I reviewed our code, upgraded jQuery little by little, fixing things as they broke
For some reason, you mentioned writing reusable functions as a benefit of using jQuery. Using jQuery guarantees that your code will not be reusable. Sure, you can refuse to upgrade and artificially extend the life of your code, but you'll inevitably end up maintaining multiple jQuery versions, eventually loading multiple versions of jQuery on the same page! (If you use a lot of plugins, things will go to hell even faster...)
Think I'm joking? It's such a common problem that jQuery includes features to help you cope with it's instability (see: noConflict). There are a host of tools (like jQuery Quarantine) to help you manage the hell that you created because you swallowed a load of nonsense about a crummy library back in 2006. Er, sorry, read that as "thought you could save some time by using a popular library."
You could wind up essentially rewriting it, but chances are it won't be shorter/more efficient.
That's just absurd. jQuery primarily consists of functions that are common across browsers, wrapped in an incredibly inefficient library. It's primary (and only useful) feature was its selector engine, but that hasn't been an advantage for a very long time now. Once you learn more about JavaScript and the DOM, you'll realize how little jQuery actually offers you.
This is all off-topic. My original point was that jQuery Mobile is best avoided as it's too slow. (No surprise, no one is willing to defend THAT mess!) This turned in to "defend jQuery against the unbelievers!" thread, which I'm rapidly losing interest in.