[spiderplant]

We've actually had a similar problem, but for us the rain was coming into the cavity via the windows. It's a modern (1999) stone-built house. They diagnosed the drip trays under the windows were either not fitted correctly or cracked. They ended up taking the front leaf off the house to replace the trays. Every stone is unique, so they all had to be laid out on the lawn to keep them in the right order. It was a massive job. This was all done under the NHBC warranty, so cost us nothing but inconvenience.

It wasn't entirely fixed at first attempt, but we've since had the windows resealed and so far so good. It's south-facing so only leaked in the rare combination of prolonged rain and a southerly wind.

Sounds like the proposed solution for you is a similar tray under the roof. I can see how that would work, but I think I'd get a third professional opinion.