Determinism is related to purity ("functional programming"), but is a weaker notion. For example, this function

    var c = 0
    function deterministic_not_pure() {
        return c++
    }

is not pure, and you can't reason over it equationally. But it is deterministic: if I make a series of calls to this function in my browser, and you make the same series of calls in your browser, and someone else emulates the code using a Turing machine made of rocks in the desert [1] we will all see the same results. (And consequently, there is nothing that such code can do to learn about its environment!)

As far as I know everything in Javascript-the-language has this property! [2] (Equally, so does WebAssembly). And most of the browser's APIs do as well, though the sheer size of that surface means that any attempt to fix the rare cracks will still be daunting. The vast majority of information that browser JS gets is either through events (i.e. function arguments) and I think most of the rest could be safely forced to be deterministic during the execution of a JS function.

It's still obviously possible for these events to leak timing information about the execution of JS, but I think that can be fought. For example, you could delay the delivery of every event by the number of nanoseconds that JS has executed on the page so far, though of course that naive approach will gradually degrade in performance. But I think for example doing that until you have 10ms of delay and then resetting would probably make timing attacks totally impractical.

All of this needs way more thought, and I'm not at all suggesting that it's easy. But browser JS is closer to an actually-securable-against-side-channels state than almost any other widely deployed platform, and I think it's sad for the people that control the platform (which is, let's face it, basically the Chrome team) to throw it away because they love performance.now() and SharedArrayBuffer.

[1] https://xkcd.com/505/

[2] Well, ECMAScript probably doesn't specify a random number generator, for example. But it could.