OK, some more searching and I found enough hints‡ to find something that works, whether it is officially documented this way, I have no idea. I would love to know where the documentation might be.
https://docs.google.com/spreadsheets/d/ID/export?format=tsv&gid=GID
The ID is what you would expect, the ID of the sheet's key from the URL. But the other surprising thing with Google Sheets, is that the GID no longer predictably starts at 0 for the first sheet, and increments for subsequent sheets. Whether this is supposed to add security or is an accident of the implementation, or an especially useful design for some internal reason, I do not know... but the GID numbers on a 7 sheet Sheet I just uploaded from XLS format are all large, and appear random.
While a particular sheet from a given spreadsheet may easily be determined from the URL displayed while viewing that Sheet in the browser, it looks like the knowledge that "all Sheets in this folder contain only one sheet" is insufficient to be able to download them, and the ability to "download the first sheet from each of these Sheets" is eliminated, unless the GIDs can be determined.
In fact, it seems my whole problem was the assumption of the first gid being zero, which it always has been for old Sheets... the URL form in this answer seems to be closest to what is used for editing new Sheets in the browser, but both forms of URL in the original question also work, if the proper GID is used.
Bonus points for a technique to determine the GID... based on what I thought I knew about old Sheets, I was planning for a system of 8000+ Sheets, with 4-6 sheets each, and thinking I could download them, one sheet of each Sheet at a time, by iterating through low-numbered GID values. KaBLAM goes that design, with the non-predictable GID values!
‡ The hints start here (but this solution seems excessively lengthy) and proceed to the here and here.
Best Answer
The only sure-fire solution I've found, after days of scouring the web and trying everything I could find, is to use ImportXML. It appears to work consistently every time, while importhtml() fails quite frequently. Here is the tutorial I followed, it's actually incredibly easy to use and works far better!