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