So what you're saying is that the usb modem itself has a web interface that allows for SMS reading, amongst other stuff?
Interesting to head that some cell modems have a web interface. The one I'm using (Telit via SixFab) does not, only an AT interface. At least as far as I've found in the documentation so far.
I'm not even exactly sure how SMS are received at all on various devices. Do they constantly listen on some kind of GSM stream? Is that the same as receiving a call?
Yes. Cell phones are in constant low-level contact with the nearest cell tower, mostly to tell the tower "I am here and I will be able to hear a broadcast". Towers then regularly send out a broadcast, which says "Hi, I have some traffic for 123-456-7890, if you hear this please come online". That traffic could then become a SMS, or a phone call, or just a request for "didn't hear from you in a while, tell me that you're still around".
And then who exactly accepts the message - a modem, some hardware down the line, is there a queue, are messages stored in a hardware buffer etc... When I don't know details like that, choosing anything is very difficult.
As far as I can tell from looking at the interface, the cell "phone" does have a persistent memory buffer for multiple SMS messages, and there seem to be commands to read and delete from these buffers. I didn't go into great detail on that: Since I need to set up IP connectivity over the cell modem anyway, I decided to not bother with SMS, but instead use traditional e-mail (SMTP and all that) once the IP connection is up.
You can say it's for personal use, I just want to check on those messages from time to time, remotely, maybe write a simple script to do email delivery to my inbox.
That might be easier to do if the cell phone provider has a service that allows e-mail based SMSes. In reality, SMS come from the cell provider (In the US, typically AT&T, Verizon and T-Mobile), and are held in their cloud during transit. These providers know how to interface to other services, right from their cloud. I know that in the US, the Twilio e-mail service had some SMS capability in its sendgrid product, but I never looked into the details. You might not need to use a cell phone or modem at all to integrate SMS.