Basically the title of the thread. Currently other people can only play my (private access) game in their browsers, but if I test making the project downloadable, it only downloads the source code.

I have not been able to understand your question well.

If you want your game to be playable in the browser and at the same time, downloadable. You need to upload two versions. indicate that one will be played online on itch.io and mark the other as a download file.

Now, an HTML5 game is not an executable, but code, which must be executed on a server (at least in a browser, but it depends what you have inside the HTML files.).
If a person downloads that, he's going to download the HTML code, because that's the game and he probably can't play it because a normal PC doesn't work as a server.

If you want your game to be an executable and not HTML code, it is best to create your games in an engine that allows you to export to different target.

For example, creating your game with Godot.
You export to an executable binary so it can be downloaded and played on Windows computers.
You export to HTML to upload it to a server like itch so it can be played online.