I think they are great! Especially in combination with pattern matching (https://github.com/tc39/proposal-pattern-matching), or when using JSX.

Do you have any insight into how the TC39 process works, and how one might drive this work forwards? I feel like expression-orientation is the main thing missing from JavaScript at this point, and I'd love to contribute. But it's not at all clear to me how to actually do so.

The TC39 process is well-documented, you can check out the stages and guidance for providing input here: https://tc39.es/process-document/

The do expression proposal is currently stage 1, so its very early. You can check out the issue tracker there to see some of the related discussion. Standards work can be deceptively hard, even for simple things. I think the do expression proposal is a good example of that.

Edit: forgot to link to the issue tracker I referenced above https://github.com/tc39/proposal-do-expressions/issues

Do you have a link to the proposal?

https://github.com/tc39/proposal-do-expressions

Allows you to contain temporary variables to where they are needed, rather than having them remain active through the remainder of the current scope. You could do that with immediately invoked function expressions, but they are too verbose to be really viable for this. You could also use extra functions, but extra functions don't always make sense. Currently I'm frequently doing this:

Without do expression:

    let result;
    {
        let tmp = 123;

        result = tmp * 2;
    }

With do expression it would become this:

    let result = do {
        let tmp = 123;
     
        tmp * 2;
    }