Just a minor correction (and sorry if I misread anyone here), but AGA does possess a 24-bit color palette, which is supported by the Indivision. Some scandoublers don't, and the loss of definition is obvious under certain circumstances. (The color sphere test images included with PPaint are an easy way to check both the scandoubler, and LCD for complete color representation.)
The Indivision is a scandoubler/flickerfixer, which in simple terms means it modifies the Amiga native video signal to something "understandable" by modern displays. It doesn't really possess the same functionality as an RTG video card. In its intended arena, it scores some major points over older scandoublers--it's highly configurable (a huge plus for good compatibility with various displays) and two, its DVI output renders an absolutely stunning and flawless static image quality (satisfactory scrolling in games requires careful configuration *and* the "right" monitor).
Essentially, it's a nice tool for having the flexibility to use almost any modern monitor. For gaming perfection, it's arguable that something like a 50Hz capable Sony broadcast CRT is ideal, but of course that's also lacking for desktop use.
FWIW, I felt the MK2 Indivision was a huge improvement over its predecessor, and over other scandoublers I've had in the past. The lack of a perfectly synchronized 50Hz mode is the only fly in the ointment as far as I'm concerned. I do agree, an Amiga with some sort of video card is really nice (something I wouldn't do without, either). I didn't experience heat or connectivity issues here. I would consider using some form of a flash drive to keep the heat in check, and thoroughly cleaning Lisa's contacts and try installing again. (I did experience both heat, and contact issues with the original Indivision 1200, and eventually sold it with no regrets).
Finally, the choice of LCD makes quite a difference as to what kind of experience you might have using the Indivision. I've used various Eizo and Samsung units here with good luck, some of them sync down to 50 Hz via DVI or (in every case) HDMI. This is somewhat important, since resorting to using the analog signal produces a small but obvious decrease in picture quality.