Sorry to say... like many things.. it depends
I'll comment in case no one else does... I'm not a gamer, so can't comment on that aspect at all.
And some folks have talked about streaming on low-end systems. But that takes a lot of knowledge to optimize you setup. Video encoding is hard work for a PC. And think about how long you want this to work. Getting something mid-range to work today isn't that hard/expensive. But will that still handle encoding duties in a few years?
Now granted, I've not looked into a dedicated streaming PC for your use case. I did just buy a dedicated streaming PC for House of Worship streaming (so MUCH lower CPU/GPU demands than gaming, but as you aren't gaming on this PC probably not that dissimilar (but I'm only guessing) I'm using a USB webcam for now, but switching to NDI PTZ camera very soon (pickup up today, hopefully). So streaming computer has decoding of incoming video source at 30fps (your system would have 2 streams to deal with, and at what refresh and fps?) For me, planning to get 4-5 years of use on machine considered a 6-core/12-thread CPU, GTX 1650 Super GPU based system to be a minimum (system would also be backup Video editing PC using DaVinci Resolve). I preferred and ended up with 8c/16t CPU. I also got a 1660 Super but only 'cuz that was the closest available SKU above a 1650Super (Turing based NVENC)
I can't comment on NDI vs capture cards. each has its own Pros/Cons including cost, latency, & complexity
As for Internet - your gaming rigs are on same local network right? In which case NDI between gaming PCs and dedicated streaming PC has nothing to do with Internet, only local traffic. Beware that a 5mb/s UL rate is good for 1080p 30fps at 4000kbps bitrate (if that) and only then if you are careful with whatever else my use Internet (you'll see all kinds of comments about how much spare bandwidth you'll need. In reality, not that much spare *IF* you know exactly what other data traffic is going on, how to control it, etc... 99.99+% of home users are nowhere near that sophisticated, hence rationale for large buffer between UL rate and streaming video bitrate. Also, Internet bandwidth isn't static, just as highway traffic isn't, which is another reason to not try and max out you UL rate
