Well unsurprisingly the cheapass upscalers can ruin your picture in the conversion process (especially if you use composite video... I used to use one of those too and it was trash). You do get what you pay for, so there are better upscalers that are excellent for converting those legacy systems to output HDMI.

PS2 and Wii have readily available Y/Pb/Pr component cables on the market, so some of those in conjunction with a scaler like https://www.amazon.com/gp/product/B017LUXQB6 can actually get you a pretty high-quality HDMI picture that you can then feed into just about any capture card.

I know it's not ideal being forced to pay for extra gadgets, and since I've personally never used that model of capture card, I can't say what the comparison will be like for you, but I'm pretty happy using component video and an external converter. I can record my PS2 and Wii, and by extension PS1 and any GCN games that support progressive scan, with an HD60 Pro and it's been solid. I suspect the HD60 or HD60 S would get good mileage out of a similar setup too.

For other consoles, My Life in Gaming has a series (which I believe is still ongoing) of videos demonstrating--in absurd detail--how to get the best HD picture from older consoles. https://www.youtube.com/playlist?list=PLTNBVisVMbSRlu45wgWo332Ke64e1o05m for the full playlist, which right now has a bit of everything from the NES and Master System up to the PS2.