Sweet, thanks for the context, detail, and hilarious-as-long-as-it-doesn't-happen-to-me backstory.
The 1756-L61 is old enough not to have the useful dot-matrix scrolling display (but the 1756-ENBT has one !), but rather just indicator LEDs.
The indicator to look at on that 1756-L61 is the "OK" LED on the lower right. If it's solid green there's an operable program (even the default that is loaded after a power loss without a battery).
Solid red means an unrecoverable fault; you have to cycle power.
Flashing red means a recoverable fault, one that you can clear by going online or with a keyswitch cycle.
That version was unpleasantly buggy, to the point where it was removed from the webserver and retired. In fact, RA is at the point with the 1756-L61 that they only publicly allow you to download the last available v20.x version (v20.019).
I'm not saying that for sure that you're encountering a bug, or if there's something wrong with your network or computer or configuration.
But if I had the chance, I would load RSLogix 5000 version 20.06 on my PC, and version 20.019 on the controller, and then re-download the program and re-burn the CF card image.
Because the CF card image includes the firmware and the user program, any time you power cycle that old image will be re-loaded.
If you're not up for that, at least get a look at the OK Led when this fault happens. In RSLogix 5000, look at the predicted memory usage and see if the controller is very close to full.
The other tests I would do is try to go online when the controller is in PROG mode, and/or connect it to an Ethernet network where it can't talk to the HMI and drives, and try to go online with it under those conditions.