The only reason I mention CPU manufacturer is mainly because of the motherboards themselves. AVerMedia capture cards rely on specific USB controllers that are found in all Intel chipsets but not AMD. You can probably get them to work with AMD, but I didn't have any luck so I ended up getting rid of all but one of my AVerMedia capture cards because I switched to AMD. Elgato also had issues with AMD chipsets in their older versions, but they fixed this and if you happen to get an Elgato card that doesn't work with AMD they'll replace it with one that does if you open a ticket with them (I've done this twice). Since you're using Intel, the brand doesn't matter though.
I've used both the HD60 Pro and HD60 S and they are identical in terms of quality. The HD60 Pro does have the H.264 encoding but only if you use the Elgato software and it's not as good as x264 at Twitch's 6000 bitrate limit. I use the HD60 S with my mITX streaming PC and prefer the USB capture cards over the PCIe ones mostly because they're portable so for instances like yours where you'll be streaming consoles you don't need to exclusively stream near your PC if you happen to have a decent laptop (or plan on getting one in the future).
Both AVerMedia and Elgato are really good options and both are roughly the same price. If I didn't switch to AMD I would have stuck with my AVerMedia GC550, I liked the quality and settings a little better than the Elgato HD60 S. Both do 1080p at 60FPS no problem, but lately I've been downscaling my video to 720p because I lowered my bitrate to 4000 since, like you said, you don't get transcoding on your Twitch channel until you're partnered (and sometimes if you're affiliated depending the viewer count I think).
If you're looking to save money, check eBay for something used. I've gotten all but one of my capture cards on eBay and so far not a single one has had any issues. I've been able to resell the cards on eBay for roughly the same price I paid for them 6+ months later so I could use the money to upgrade to a better card. The most common thing I've noticed with eBay is that the boxes usually don't have a video cable like if you bought them new, but if you have extra HDMI cables then it's worth it to save some money.
I can't believe I wasted almost $100 on the Noctua and Cryorig (and brackets) when the Stealth is going for under $15 on Amazon. Not to mention installing the Noctua and Cryorig are extremely hard, for my case I have to pull out the motherboard to install, remove, and re-install them and then you need to balance the HSF and motherboard to get the bracket lined up correctly. It's impossible to use the thermal pads with them because they slide out every time but the Stealth can be installed, removed, and re-installed without having to remove the motherboard.
Aslong as i know AMD partnered with coolermaster and that is why the coolers are so good. Also this is the case atleast with the 1700 cooler, it is not only completely aluminium, but instead it has a vapor chamber