I have heard in the past that programs are designed not to trigger an alarm for something like this because they know that raising an arm quickly simply to look at the computer happens on the majority of dives.You shouldn't have a problem triggering the ascent rate alarm: move your arm up as you go up.
I don't think you need to actually see some of the alarms or warnings to recognize them when they are happening. I once surfaced at the end of a dive during a multidive day, and someone who was with me on that dive (but had been diving with others earlier) said his computer was doing something weird. During the dive it had given him some kind of message that I don't recall now, and it had started counting up minutes. I didn't know that computer at all, but I immediately understood that he had gone into decompression, and he was getting an increasing time to surface because of the now required decompression stop. I immediately took him back down, and we did an extended safety stop. When we got back to the surface, I had him go on oxygen for a while and monitored him.
His original diver training was table based, so he had not had any training on computers in general. He did not know that computers in general will do that. If he had looked at his manual to see how his computer did it, he would have recognized it without having seen it before.
It sounds as if you have enough understanding of what computers do to look up those details in your manual. You are way ahead of people who just buy computers, don't understand what they do in general, and don't look up the specifics for their specific computer.