In a conversational setting, probably 80% of my responding to the other person starts with "아~" in a very specific tone tailored to the situation.
"아~ 네 알겠어요"/"아~ 알았어" - usually when the other person has asked me to carry out some instructions
"아!"/"아~!" - when the other person was explaining a difficult concept and then it finally clicks for you, "I get it!"
As you mentioned "알았어" just by itself has the potential to be interpreted as rude in a certain tone so I would always add a short or long 아~ at the beginning just to make it clear that I'm saying "I see, I understand now" instead of "I get it, stop talking to me"