Let's be VERY clear here. Saving "via the web browser of your choosing, anywhere, anytime" is not gone. It just relies upon the "default download saver".
This saver -- which is built into the TW core code... NO plugins or external programs needed -- DOES work in almost all browsers. It does this by triggering the browser's standard "download a file" handling to deliver the changed TiddlyWiki content to your filesystem, **even when working with local files**.
The exact interaction "flow" varies a bit depending upon your system and browser setup. By default, most browsers save files into your "Download" directory and, if the filename is already in use, will either prompt for overwrite, request a different filename, or automatically add "(n)" suffix to the filename to avoid the conflict.
Many browsers also let you enable "Always ask for location before saving", which will display the local system's "Save As" dialog to let you choose the location (and filename) to be saved. In this dialog, some browsers default to the "Download" directory, but most will default to the *current working directory* (i.e., the folder you were in when you opened your TW).
So, for example, let's say I'm working with a TW file named "index.html". With the download saver (setup as above), when I invoke the Save File command in TW, a dialog box pops up, pointing at the current directory. In the file list, I can see the file I am working on (e.g., "index.html"). On my platform (Win10), the dialog automatically suggests a new filename of "index.html (1)". But, I want to overwrite the original file, so I select "index.html" from the file list. Then I press the "Save" button and I'm done.
In summary, the sequence is:
1) press Save File button in TW
2) select "index.html" in the filelist
3) press Save button in file dialog
Note: when working on complex changes to a TW document, I often save "checkpoints" along the way, allowing the download saver to add the "(n)" suffix to save files, before continuing to edit my document. This gives me an easy way to back out any changes that break things. When I am satisfied that my changes are good, I save back to the original filename (without the "(n)" suffix) and delete the "checkpoint" files to clean up. The actions need to save are even more trivial than above...
1) press Save File button in TW
2) press Save button in file dialog
I even use this process when starting new documents *directly from TiddlyWiki.com*. I simply go to
https://tiddlywiki.com/empty.html and start editing (creating new tiddlers, configuring Control Panel options, etc.) When I am ready, I invoke the download saver to save the file (with my changes) on my local filesystem. Then I open that local file and continue as above.