"What do the pros use to provide synchronization to old TR808s, 909s, TB303, and other ancient drum machines?"

If you encounter a drum machine with MIDI, no matter how old, there is a good chance it will respond to MIDI clock and Song Position Pointer. Most sequencers can output this type of information with some simple preferences and selections on your part. Occasionally we run into someone who wants to slave one of these directly to a tape machine. Though it can sometimes be done if you have the right equipment, we don't recommend it. Get yourself a sequencer and run everything from that. Slave the sequencer to the tape machine by whatever means necessary (usually a MIDI interface with SMPTE built in if it's a computer based sequencer) and go from there. If you only have a hardware sequencer such as those built in to many keyboards you may still be at an impasse because these quite often do not chase Time Code and other synchronization protocols commonly used with tape machines. Some digital tape machines will output MTC, which gets you one step closer, but you still need a way to convert that to MIDI Clock and Song Pointer. Computer based sequencers are generally much more adept at doing this.

Some very old drum machines do not have MIDI. In most cases they will have a type of pulse or clock input that can be used as a crude form of synchronization. They work by sensing a pulse or clock signal and using that to trigger incremental advances through the pattern/song/sequence at hand. The number of pulses required to move the sequence one quarter note will vary from unit to unit. This is known as Pulses Per Quarter Note, or PPQN. If you don't know what your drum machine requires it is relatively easy to find out just by experimentation. The problem is that it is not unusual for some pulses to get 'lost' by the machine and sooner or later they generally fall out of sync, with no way to recover. Of much more difficulty is finding a device that will output these types of analog clock pulses. The devices that generate these disappeared not long after MIDI took over. Industrious types have found numerous ways to work around this problem (including a few who use synthesizers to generate pulses), but all in all most pros just get what they need into a DAW and then edit it from there. You won't find too many people (at least those who value their time) messing around with trying to keep these old beasts synchronized to a complex system.