Again, you seem to be using a bunch of buzzwords.
Framework 16 is currently only available with AMD APUs/CPUs. Thunderbolt is an Intel/Apple co-developed and licensed technology/certification. While AMD laptops can get certified to use/support/offer Thunderbolt, in order to do so, they have to give money to Intel (one of their prime competitors) for Intel to thoroughly test and certify that AMD’s product meets Intel’s standards. It happens very rarely. As of the time of writing, Framework 16 is not* Thunderbolt certified.
Thunderbolt 3 uses a USB-C connector, but is not developed or maintained by the USB-IF (the group that does maintain and stupidly name USB standards).
Intel supports USB data traffic on Thunderbolt connections, as well as power passthrough, IP traffic, PCIe passthrough, and video signals.
USB 3.2 Gen2(x2) supports USB data traffic, power passthrough, and video signals.
The biggest difference is Intel requires that all Thunderbolt devices support all of the things, while USB says that pretty much everything is optional. It doesn’t have to support charging to be USB, it doesn’t have to support data to be USB (USB charging cables), it doesn’t have to support video, etc. All of this is on display with the Framework 16. Some ports don’t support video output, some don’t support charging, and if you use the port on the back of the dGPU, while it does support data, it’s limited to USB 2.0 speeds (even though the DP/HDMI signals being passed though it are likely 20-30Gbps bandwidth).
But as previously mentioned, certifying every laptop to support Thunderbolt is expensive, so there weren’t as many devices getting certified as Intel would like. Fewer certified laptops means fewer potential purchasers of Thunderbolt products like docks, screens, E-GPUs, or storage products (and Intel gets royalties from each product sold). So Intel cracked open Thunderbolt just a little and told the USB-IF that they could have/use Thunderbolt 3 for free right around the time that they were introducing Thunderbolt 4.
USB-IF then introduced USB4, which is mostly just a rebadge of Thunderbolt 3! But they did it the way they’ve introduced everything; by making nearly everything about Thunderbolt optional. (*So Framework 16 is NOT Thunderbolt certified, but some/many Thunderbolt 3 products are likely to work in the 2 ports closest to the screen.)
But this long, and probably boring story finally comes back to what you asked.
You can’t receive the video signals being output from an output device (like a Thunderbolt/USB4 enabled laptop) into another output device (like the video output on another laptop). While many things with Thunderbolt and USB-C are bidirectional, like USB data speeds or IP traffic, many others aren’t. Thunderbolt requires laptops are able to receive 100W (or up to 240W) to charge, they’re only required to output 15W. While they’re required to support HDMI/DP video content output, they aren’t required (and likely can’t) receive it.
That’s where that capture device comes in. While you can’t send the video signal back in to an output, a capture device is an input. Once the capture device receives the video stream data, it can encode it into a protocol/data stream that Thunderbolt/USB is capable of receiving; usually USB data traffic that gets treated as a camera, because that’s usually the only incoming video stream data that devices are made to expect.
There’s no technical reason I can think of that would prevent someone from sending the data over another supported bidirectional data type, like IP with Thunderbolt (many professional video production businesses send live video over dedicated, high-speed, but otherwise standard networks), but then there’s additional decoding overhead that I suspect would add latency.
As for USB-C, it’s just a connector and sadly does not enforce or require any specific capabilities (other than hopefully not being intentionally malicious). It can be used as a low-end 5v-only charging connector with no data connection, USB 2.0 data device (yay 480Mbps!), USB 3.0/3.1 Gen 1/3.2 Gen1x1/SuperSpeed at 5Gbps, USB 3.2 Gen 2x1 (SuperSpeed+)/USB 3.2 Gen 1×2(SuperSpeed) at 10Gbps, USB 3.2 Gen 2×2 (SuperSpeed+)/USB4 Gen 2x2 at 20Gbps, USB4 Gen3x2 at 40Gbps, video over DP alt-mode, video over HDMI alt-mode, support USB-PD at up to 60W/100W/240W, or PCIe passthrough/tunneling.
As for powering a Framework 13 mobo… I don’t know the minimum power draw requirements for the 13 (the 16 “requires” a 60W USB-PD charger minimum per Framework docs), but if the battery were connected, sort of as a built-in UPS, and you weren’t running a high draw compute load 24/7, you’d likely be able to charge it slowly. That being said, I’ve skimmed the thread of folks complaining about low USB power output on the Framework 16, and think Framework is likely following the USB standard’s minimum (7.5W) instead of the Thunderbolt standard’s minimum (15W). So a very slow charge.