xbox controller blinking?
change its batteries or plug it into a usb charger or brick (non data device or using a charge only cable)

It has fresh batteries, but I can try again.

And I played for like an hour with no problems after posting yesterday with no other bluetooth controllers connected.

The fact that your controller is starting to have issues is a bit suspicious.
Still, this is relatively easy to diagnose, either Steam Link, the controller or the device.

At first glance, if the controller is blinking, it is stopping being recognized by the device it is connected to. If it was having translation problems with Steam Link, you would see the light on, but no input occurring.

Firstly, I would suggest you try to use the controller directly on your device, without steam linking anywhere. Then try the same, but connect a cable.
If issues arise if connected by Bluetooth, but not by cable, then most likely the problem is with the device or Bluetooth dongle that is listening to the controller.
If the controller works fine by both methods and not while Steam Linking, then Steam link is the issue.

Recently, I got a controller for myself and I was having pretty similar issues.
What happened to me was that my device wasn't using the right drivers for my Bluetooth dongle. Do NOT trust windows to update those for you. It doesn't work.

You need to check for the model of your Bluetooth dongle on the website of respective brand for the most recent version. Compare that number with the drivers you have installed to see if they are properly updated.
You can check all information for your dongle on "Manage devices" menu, on your PC.

Through that same menu, in case the numbers do not correspond, you uninstall older drivers, then manually install new ones.
This instantly solved all my issues with my Xbox controller.

Controllers are usually not very demanding in terms of power, in my experience, they only die when batteries are really depleted. I find it very unlikely it is a power issue.
Still, when trying to diagnose this myself, I found some people who were complaining about their controller not working very well with rechargeable batteries. You can try to use different ones, accordingly.

I appreciate the attempt to help. I'm trying to decide how much is actionable in my scenario, though. Some key points.

1) The controller works perfectly, for hours, if I do not connect another bluetooth controller at the same time.
2) Connecting another bluetooth controller results in the connection problems within minutes/seconds.
3) There is no dongle involved.
4) I do not have bluetooth functionality on my actual PC in order to try it there.
5) I could try using it wired to see if that will make a difference. I suspect that it WOULD make difference. But the entire point of buying this controller (for me) was to have a quality controller, with a solid wireless connection, that does not occupy one of the USB slots on my Steam Link.

I have seen elsewhere that you can force the Steam Link to load an old build of the firmware. If I get the time, I will likely try going back a few revisions and seeing if that fixes the issue for me.

I see. You are connecting your controllers on your physical Steam Link then, if you are concerned about the USB ports.
Seeing that you are connecting them to the Clients' side, I doubt that it would be a drivers issue on Host side, since that is basically only receiving the inputs and information that controllers are present.

I can only imagine that there could be some kind of conflict between controllers if they are all trying to access the same Bluetooth, which shouldn't really happen. I'm not really familiar with Steam Link hardware but you can try and see in the options if there's anything related to multiple controllers or accepting multiple inputs. Assuming that it could have been updated, options can be reset sometimes.

Now there's multiple things I could think of if options do not work, nor reversing firmware.

Probably the easiest would be to try to connect one or both of your controller to the TV instead. TVs nowadays come with USB ports and Bluetooth. The question is if Steam Link will recognize the controllers if not connected directly.
The second simplest would be to acquire an USB extension and connect an extra dongle there.
The last resort would be to forfeit physical steam link as a whole, if you have a smart TV. If so, you can just install purely digital Steam Link and connect your controllers to the TV instead.

Yeah, no smart TV here. The value proposition of the Steam Link hardware is definitely a major issue for me as well.

Sad to report that reverting to older versions of the firmware did not seem to fix my issue, so I'm left baffled still. The problematic controller still works for hours if it's the only bluetooth device that is connected. Multiple other bluetooth controllers also work fine together on the Steam Link. Very strange.