I can't think of anything that would directly match your requirements other than using another uC programmed to act as if it's a soundchip.
SID... has the waveforms, at least most of them, but doesn't have the envelopes, and probably too expensive these days
AY3 / YM2149 ... has the envelope (of a sort), but only squares and noise
SN7 is essentially the same thing just with certain subtle differences, none of which are relevant, other than the loss of the envelope.
Nintendo chips are themselves also CPUs, so a bit beyond the scope.
Various Yamaha FM chips come close, particularly the OPL3 may offer a wide enough range of timbres to more or less cover your requirements - or the simpler OPL2 or OPLL may yet be flexible enough to displace that requirement; alternatively the OPN series combine an AY3 core with various levels of vaguely OPL-ish FM and/or (AD)PCM (the latter needs a source of sample data of course) which can cover a lot of the bases, and built-in ADSR is a given for all of the FM engines (albeit only programmable for a single voice with the OPLL). Again, the issue is availability and cost. I wouldn't even want to guess how much any one chip may set you back.
About the closest direct hit is the Atari POKEY, it may not have those exact waveforms but it has a variety of interesting algorithmic ones which are probably fairly close. Though admittedly IDK if it does hardware enveloping so it may still be worse than the SID, and it's probably no cheaper.
If all you want is some bloops with at least 2-channel polyphony, I'd suggest looking for an AY3 or its Yamaha clone (especially, if you don't need the IO ports, the cut-down later models, as those are rather more compact and thus more suitable for a dinky microcontroller project, plus probably less sought after/collectable overall), or the Texas Instruments soundalike as featured in the BBC Micro, Mockingbird PC board, cloned in the various Sega VDPs etc, which is even smaller and probably cheaper even as NOS. You can get some reasonable sound out of them, as demonstrated by the systems they were installed in (the SN often doing about as well as the AY/YM because it has a few nonobvious advantages that compensate for e.g. lack of hardware envelope), and if you figure you can knock together a replayer routine for your chosen controller there's an absolutely humongous library of ready-to-run music capture dumps out there waiting to be plundered. The cleverer ones for the AY include stuff like SIDvoice, syncbuzzer, digidrum, even samples, etc which make use of the envelope, or note retriggering, so on and so forth to e.g. emulate the SID triangle/saw waves, make a fair stab at a sine, adjust the duty cycle of the squarewave into something more like a Nintendo pulsewave, create a reasonable set of 303 drums using square-noise combos... and use volume control abusing sample replay to emulate everything else. Potentially computationally expensive, but at the end of the day all the replayer really has to do is decompress the packed file on the fly (if it's not been pre-unpacked) and then fling the data contained within at the relevant registers in accordance with the internal timestamps.