After building a prototype of the first circuit, I found a few things that weren't very good: First, the speed and weight pots interacted significantly due to a flaw in my design. I'll call it a brain fart. The adjustment range was pretty much unusable as well, and the output waveform was pretty funky.
Since it kind of worked, I did some further noodling and came up with the tweaks shown in the circuit above. While I haven't built it in the physical world yet, I've modeled it in LT Spice and it looks pretty good. The adjustment range for the speed control doesn't go as slowly as I'd like (about 13 WPM), but I seldom send any slower than that anyway, and the mid-scale speed of about 25 WPM suits me just fine. If I find that I need it to go slower, I can always add a "range" switch to add more R or C to the circuit - not a big deal.
With any luck, I'll have time to build this tomorrow and see if it works as designed. Stay tuned!