I'm for sure no Atari guy so maybe I'm wrong - but I think our 6502 machines tend to be powered on and off rather quickly / often. But any Linux SoC thingy takes ~30 secs to come up and usually wants to be shut down cleanly. So I'm with you that I prefer clean integrated solutions I have by now come to the point to see the Linux SoC as an optional add-on.
On a Raspberry Pi you can for example run a SIO emulator and and a SSH client at the same time. But maybe you don't have much fun connecting both at the same time to the Atari. But with the Atari having an Ethernet interface with (in contrast to the Dragon Cart) it's own TCP/IP using the W5100 one can at the same time run SIO over some simple proprietary UDP protocol and run Telnet over a TCP connection. And with Telnet one logs on to the Raspberry Pi and SSHs from there to anywhere.
This setup surely blocks the single Ethernet port on the Raspberry Pi. But with Ethernet you can just connect both (the Atari and the Pi) to your LAN and have the Pi located anywhere. Or you're into WiFi, connect the Atari via a short Ethernet cable directly to the Pi and use the Pi's WiFi to connect to the net. You can even have the Pi act as WiFi bridge allowing Atari Ethernet software like the Contiki web browser to directly connect other machines on the net.
SSL/TLS is something you can get today in things like an ESP8266 / ESP32. But I'm not aware of a (free) SSH client implementation for any micro-controller platform. So from my perspective you end up with some Linux-capable SoC running OpenWRT or Raspbian.
But maybe you are rather refering to some web browser / wget running on some box you are connected to via Telnet. Then it's obvious that you can only download files to the file system on the box you are connected to via Telnet.
Even if you run the Contiki wget on your Atari you will only be able to download to a file in your Atari file system, but not to "raw blocks" on your disk. Maybe I'll provide a new wget that will be able to do the latter.