An update-- nearing the end of this saga but not entirely finished. At this point I am just updating to share info with anybody else researching this currently or down the line. Feel free to comment though and thanks to everybody in the earlier discussion phase for helping me make this happen.
3 of my 4 boxes (medium moving box size) were successfully held at Sihlpost, the 4th should come in a few days. They had them on a cart in the area right behind the counter attendants, which was not too crowded. But I would call them before trying to send many more boxes than that, as I'm not sure what they would do with say 10 or more boxes piling up. And especially, imagine if this method became popular and a few people did it around the same time.
But anyways, I rented a mobility.ch car after all (in fact I signed up for mobility while still in the US and already had my mobility RFID card-- for unlocking the car-- come to me by mail in the US), parked in front with the blue disc thing (not sure it was a blue zone but I put it there anyways). Sihlpost was not hard to navigate to from where I am. Google maps GPS made this part easy.
For picking up the boxes, be warned that the Post attendants don't necessarily speak English. Both at the small post office I addressed my boxes to and the Sihlpost. The first time I went to Sihlpost to ask about my packages after they were refused by the smaller post outlet, they had to shuffle me from counter to counter-- first to a younger guy whose English turned out to not be good enough, and then after that to a young woman who was much better. The second visit, I managed to get through the interaction in German. Not to scare anybody off from this method-- I think I would have gotten my packages either way as I had my email address on the boxes and on the USPS labels (and in their system, which presumably would pass that info along to the swiss post also), and there is always google translate for difficult communication situations, although the less prepared you are the more you will test the patience of people, and you will want to minimize that throughout an international move, because it really runs you down to feel like a net drain on society even just for a couple weeks or so, and especially as a guest in a new country.
Another thing about picking up the boxes. They aren't necessarily equipped for this. They had roll carts in the back, but I asked and they don't have them for the customers. So I had to take my boxes one by one out to the car. I left the stack unattended briefly as I did this, having done this by myself. Not a big deal, but it could affect your plans depending on the value of your goods, their weight and size, etc.
The car was a Citroen C1-- small but the boxes fit with folding down the back seats.
I did not pay customs, at least not on the 3 boxes of 4 received so far. I had "personal effects" or whatever on the USPS customs forms, and a lowish valuation. I did not fill out or include the swiss customs form 18.44 as people have done. I simply ran out of time.
Anyways, then drove to zebrabox. I was mindful of the trams thanks to the warning I got in this thread (I also tried to learn the street and tram configuration just walking around the city-- I would not necessarily recommend hopping in a car on your first day here), indeed driving was a bit fun for me as I got to drive a stickshift for the first time in many years, I made it to zebrabox and they were super helpful in getting me a XXS unit last minute before closing today (Saturday). They are just up in Oerlikon which is an easy 15 min drive away. Again Google Maps GPS with audio call outs and so on.
It will be around 44 CHF / mo, and I have to give 2 week notice. But for now it's way easier for me to just have the damn boxes out of the way. They were all almost exactly 50 lbs each, and between $180-$190 to ship. I did my first week in a hotel, next 3 weeks in an airbnb, and as soon as I have a minute to spare I will try to find a more comfortable place. I may make a final update once I finally unpack the boxes, to report on any damage, how well my packing job survived the shipment, etc.