I recently came across a schematic that showed how to build an oscillator using the NAND gates inside a 7400 chip. After poking around online, looking for the 7400 (and not some variant) I learned the LeTourneau University College of Engineering has a parts supply closet with a whole stash of them for 5¢ each. (I've known about the parts closet for quite a number of years, but only recently discovered a tall filing cabinet full of most commonly used ICs. (No NE602 or 612s, I'm afraid, but that's another post for another time.)

So, with my 7400 in hand, I was able to breadboard an oscillator using the "colorburst" crystal at 3.579 MHz. The oscillator emits a square wave at the fundamental frequency, so harmonics abound. In fact, just through playing around with another receiver, the 3rd harmonic at ~28.632 MHz is considerably stronger than the others that fall in the ham bands. (All higher harmonics are just above the ham bands, though I may be able to pull them back down with a variable capacitor in series with the crystal.) I need to do some range testing on this to see just how far I can get on the various bands.
My next trick is to build a bandpass filter network to dampen the harmonics down to legal levels. Then, I'll have a bona fide transmitter, though very, very low power. Perhaps a final amplifier is on the drawing board next.