Your scope isn't a storage scope, but then neither is my analogue HM400 scope. It's the kind of functionality that's standard with a digital scope, but an optional extra in the analogue market.
The behaviour seen around the 7~8 minute mark only works on that scope because it's storing a record of its recent measurements, and redrawing them repeatedly on the screen.
If your scope were triggering correctly, then it would display periodic waveforms correctly, as mine does. (Though long timebases would still be an oscillating dot progressing across the screen, because it can't store and redraw measurements.)
I'm not sure that I can do much more to help with sorting that scope out remotely: it seems that it needs some repair itself.
Cheap little scopes can be useful to probe around many parts of a Beeb, but take note of their analogue bandwidth before you buy anything: an analogue bandwidth of 2MHz would be enough to detect and show a 2MHz sine
wave, but the higher harmonics of a 2MHz square wave would be lost, and it's look like a 2MHz sine wave. A 200kHz scope, for example, really won't cut it. A scope with sufficient bandwidth to show the 16MHz master clock signal will probably have a fair price tag attached.
Having said all that, we can infer the correct operation (or not) of the master clock from the lower frequency clocks that are derived from it. Continuity tests between components, measuring component values, and selective replacement of suspect passive components and chips is another approach that we can follow.
Plus, even working with a DC voltmeter and a cheap logic probe, you can compare readings between your working Beeb and your faulty Beeb, to get an idea of what sort of results you should expect to see.
Have a think about how you'd like to progress, and let us know.
(P.S. It seems this thread title could apply to your Beeb and your scope!