slow rise time
You said it took a while, something observeable. So if it took a few seconds, you might see that on a voltmeter. If not, my mistake. there exists in many frequency counters a timer/counter mode of operation, but I think that's beyond the scope here.
Can you operate the DSO in X-Y mode? Turn off the time base in the X-direction.
auto-trigger
Hypothetically, I just said suppose your scope had 10 major divisions across the screen and each MAJOR division was 1 ms.
So, without a valid trigger, the scope would create a trigger because it can't trigger on slope or level or external.
Logic pulser
Let's take a look at one:
https://www.electro-tech-online.com/custompdfs/2013/03/LP425K.pdf
It's A LOT different than a function generator.
Think of it as a device that can override an input very briefly which means it can "force" an input to briefly change state.
A function generator would end up frying the components unless used at a real input.
If you, had say a simple NAND gate and you wanted to know if it were functioning, you could look at the output with a logic probe which can capture the pulse and inject one with the pulser.