The system they are proposing is very std and straightforward. It
should have been put in during construction when it was cheap and
easy.

Along with what has been suggested about fixing outside grading and
downspout issues, you need to also go outside during a heavy rain and
actually see what's going on. What you think is happening and what
really goes on can be very different. I was having a problem where
water came in one basement area during heavy rain. When I finally
went outside to check, I saw that during a heavy rain, water was
pouring out from the top of one of the 4" flex pipes that I was using
to carry the water away. So, instead of flowing into the pipe, a lot
of it was pouring out right against the foundation. I just had to
secure the pipe higher up on the downspout to solve the problem.