You're absolutely right that the only way to know what's going on is
to put on a raincoat on and take a look around the house during a
heavy rain. I just had a problem with water showing up in the
basement and couldn't figure out why. Turns out one of the 4" pipes I
had slipped over the end of the leader was leaking there big time
during heavy rain. Using a hose, you'd never see it.

Glad to hear that you fixed your basement water problem (I know you
are not the OP).

Water is very hard to predict its behavior. For example, my house has
two floors. I have downsprouts pouring water from the roof onto the
overhanging area in the first floor, and then the water is supposed to
flow into the gutter of the overhanging area of the first floor and go
down to another section of downsprout and onto the ground. That is the
design. But in a heavy rain, I saw water throwing out from the
downsprout in the second floor in such a strong force that the
downsprouts were like large water hoses shooting water a good four
feet out, and totally bypassing the gutter in the first floor and
landing on the ground at the house foundation. Of course, I had water
in my basement on that day right at the corners where the water was
shooting at. If I was not at home at that day to witness that, I would
never know the water in the basement had to do with the downsprout in
the second floor, and had nothing to do with the downsprout in the
first floor, and I would have spent time and effort trying to extend
the first floor downsprout further and further away hoping that this
would fix the problem, and I would still have water in my basement in
the next heavy rain. Really, seeing is believing.

Jay Chan