As you correctly stated, the Tek waveform rate is always dependent on what it gets from the trigger system. When in auto trigger mode, and no valid trigger is found, the update rate is driven by the auto-trigger timer (about 50ms). This is the way auto-trigger works on Tek scope, even back to the analog scope days. The idea is to give you enough of a view of what the waveform looks like so that you can properly setup a trigger. You can see if the waveform is sinusoidal. pulses, noise-like, etc., and that may help you make a better decision about what trigger settings to use.
I've seen instances where other scopes have trouble triggering on very infrequent events (like <1/sec) when in auto-trigger mode. I presume that this is due to whatever mechanism they use to switch between fast-freerun and triggered mode. I have never seen something missed with the Tek under same conditions, presumably due to the fact that the acquisition rate is always driven from the trigger circuits.
Note that on the Tek, hitting Run/Stop will pause the scope in it's tracks - thus retaining the display and all the detail in it's persistence memory. If you want a single trace, you would use the Single button.