Yeah, the GainCARD is what started the ChipAmp niche, and now there are many Gainclones on the market. My DIY designs so far have basically been Gainclones, I haven't done a full discrete amp yet, I've kind of moved along to other hobbies for a while, I'll get back to amp building eventually. When I do, I would like to do my own PCB etching too!
Actually, the LM3875 is a POWER AMP, not an OPAMP. I can't remember the distinction off the top of my head, but once a chip handles a certain power (Watts), then it's called a "PowerAmp". But, if the LM3875 isn't available, it's a moot point b/c it has been superseded by the LM3886, and this is the chip that I have been using most.
Honestly, as long as you don't need huge power requirements, the Gainclone is a heck of an amp! Likely would work great on those Tempests.
I have this kit that is a Gainclone, built it and measured it for my Solid State class a few years ago. Odd, but now it shows as discontinued. I caught it on sale for $120 delivered, and very tough to beat at that price, now it's showing as $185 and not in stock, likely better options at that price anyway. Furthermore, when I bought my kit the guy offered me a free headphone amp kit if I would post my review and measurements on his site! Score! So, I also have one of his CMOY headphone amp kits (he doesn't call it CMOY, but that's what it is, just like he doesn't call his amp a gainclone but that's what it is).
https://www.boxedkitamps.com/gobo-stereo-audio-amp-kit
And, I think this guy may be top dog as far as chip amps (gainclones) are concerned:
https://www.neurochrome.com/modulus-86-rev-2-1/
The PS isn't "wrong", and it's actually pretty common to use it for those type amp boards. It's just that there are "better" out there for this application. You may see it in a PC, but it's very common in other electronics gear, for example my 3D printer has that same style PS.
Yeah, if you treat this amp like a "radio shack novelty", then you won't be disappointed. But, I really suspect that you are going to be pleasantly surprised by this amp (even though I don't have experience with it). Here's the thing, amps aren't some magic device, the technology has been well understood for a very long time. Furthermore, you are discovering for yourself, you really don't have to spend a lot of $ to get killer audio!
Amp power supplies aren't too terribly difficult. The preferred topology is a Non-Regulated Linear PS. For the amps that I have built from the ground up, I have started with the PS design as my main goal to really understand and design correctly to fit the application. But, my designs so far have been +/- rails, I'm not certain if this amp just uses a + rail?