JPG, welcome to the forum
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You're in a Moog enthusiastic place, so most of us here are akin to tell you Moog synths are great. However, I am not as affirmative as the others posters here. Let me explain.
First of all, as Berlotti says, that most likely is an organ that you can hear in the two Black Keys' examples you gave. I'd say a Farfisa or similar. If you're after that very sound character, then maybe an organ or organ emulation is your best bet: you can get cheap vintage ones nowadays, or some modern keyboards will do the job -I've been using a Nord Electro 3 or 4 over the last years.
Secondly, that kind of organ (look for "combo organ" or "transistor organ" on the internet) are polyphonic: you can play chords. A Sub 37 (or Sub Phatty, Little Phatty, Voyager and almost evey Moog) won't allow that. They are monophonic: one note at a time. The Sub37, as said before, has the ability of playing two though. But it's kind of a trick.
That being said, there is a member here who had his band thinking his Moog Voyager Old School was an organ
And last but not least, synths are synths. I mean: you may get into synthesis easily and get pleasure quick, but you may also be overwhelmed by possibilities, numerous new concepts etc that the Sub 37 -for example- offers. However the Sub 37 has a cool one-knob-one-function interface: this is great for learning.
Don't get me wrong: all I wish is you get deep into it and enjoy turning knobs
If I was in you shoes, if you do want a Moog, then get the Sub37. For a bit less, get a SubPhatty or a used Little Phatty: they are very enjoyable too. Otherwise, I may be interested in some "educative" synths as the Arturia Minibrute or maybe the Korg MS20 (mini or kit). They happen to be cheaper, but I'm not sure they allow to send MIDI to a DAW for controlling drumset plug-ins.
I hope that helps
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