If the amp board isn't coming out of standby and sending a signal to turn on 24V, even with 24V from an alternate PSU it might not work. It could be that there was nothing wrong with the amp board (including transformer), that it just wasn't getting the power-on signal from a faulty amp board.
On the other hand there could be less smarts to it and if you supply 24V from a different PSU, it's always on, powering the amp and usable, which could waste a bit of power but hopefully not produce a perpetual buzzing sound when the source (TV?) is off or muted.
Do you have an alternate source of DC power for testing before you buy something? It probably don't need to be 24V, just for testing you could wire up a 12VDC/1A wall wart. Odds are that's enough to see if the amp board works even if it runs out of current or voltage and starts distorting at higher volumes.
Well I take that back, it depends on how the board is set up. If it has opamps before the power amplification stage and they're on power rails with linear regulators set to a voltage higher than 12V + Vf overhead of the regulator then you'd need an input voltage at least that high, but it wouldn't hurt to try (12V or whatever, 12V seems like a minimum voltage worth trying since 12V wall wart AC/DC supplies are quite common, practically everyone has some already, or a 12V car/etc battery, laptop AC/DC power brick, whatever.
Anyway, if you end up buying a 24V PSU, I don't see a need to regulate down the 5V from that when your original board seems to still do 5V, or was there 5VSB on the small transformer but the main 5V rail was on the transformer you ripped out?
If the old board provides all the 5V power needed still, just wire the 24V to it and let it supply 5V.
The 24V adapter you linked to on Amazon is probably a cheap design with dirty (high ripple) output and short lived. It is possible you could improve it by cracking it open (as non-destructively as possible), swapping in better, slightly larger capacitance major brand low ESR capacitor(s) and drilling some vent holes in the casing, small enough to not be an electrical hazard.
Even so the easier route would be choose a major brand 24V PSU instead, or if on a tight budget, accept a little lower peak amp power by choosing a high end (gaming, etc) laptop AC/DC power brick that is a little lower DC voltage, I mean the OEM major brand adapter for it instead of a generic aftermarket adapter.
Good pictures of the front and back of the amp board might help... or they might not, hard to say but that might help determine if there are things like the linear regulator(s) for opamps as mentioned above, unless they used tiny TO92 package or small package surface mount regulators that we can't ID from a top down picture.
Once you get power to the amp board and see what you're dealing with, such things like input power to opamps, if present, can be measured with a multimeter.
Now I feel totally bad about it. I need to find a replacement transformer.
