Two main things I usually get people to explore when contemplating cheap capture cards, especially if 1080p30 is an option.
1) Chinese options. You find these on the usual bangood, alibaba, dealextreme... type sites. They are quite nice, tend to top out at 1080p30 but price wise they are great. That said I had a quick look and there are not so many any more. Not for sale but an idea of what is out there
2) Older black magic stuff. 1080p in professional video world today is so old hat it is not funny (even 4K in some circles will get you laughed at) and thus if you can find one the older pro cards here do exceptionally well. You might need to find one of those Chinese HDMI "splitters" that just so happens to remove HDCP (don't know what the Switch does these days for this) but probably worth having one of those anyway as it means you don't have any lag issues trying to both capture and play at the same time. https://www.amazon.de/Intensity-Pro-Video-PCIe-Card/dp/B001CN9GEA
for an example of such a thing. 90 or so Euros is a bit steep so I imagine some people realised what a good deal they were (they were once possibly half that).
Avoid anything with USB2.0 like the plague -- they are good for stills only, and even that is debatable.
USB 3.0 stuff I have yet to be properly sold on as well as the internal card stuff but it can do it. Not sure what the sub 100 Euro stuff is like and reviews are a bit thin on detail worth noting here -- most people doing reviews here seem to want fancy software to hold their hand when I want the opposite and for it to get out my way while it speaks to something nice like virtualdub or something I can similarly fiddle with the options in.
As for streaming function then that is a function of the PC it is running on -- any random game capture box with a streaming option is just one that has the ability to save your login details really. There can be a bit more to it but streaming it not a big deal, unless you want to do fancy editing, cutaways, source swapping and whatever else -- live editing is hard.