The issue with rain in F1 is the flat floor and plank (its an issue in a number of other areas too!).
F1 cars have a massive, flat bottom that means they float, and then a long plank down the middle of them, mm's above the ground (or scraping it, if your RBR) that acts like a rudder. It means that as soon as there's a few mm's of standing water the floor of the cars start defining their direction, rather than the drivers steering inputs.
The plank has been there since the German GP of 1994 - it's not just it being there, it's the combination of flat floor + plank + ride heights being set veeeeeeery close to to the edge + banning of any way to alter the ride height while out on track to take account of conditions.
Allow active ride (perhaps a standard system, though I'd rather it not be) so they can alter the ground clearance to suit the conditions and a flat bottom car with a plank nailed to it becomes less of a problem. Of course the real solution is....
So the solution is to remove the plank and replace it with stategically placed skid blocks, and then remove the requirement for flat floors. Simple. Ground effect, faster cars able to follow each other, and an ability to race in more than a light drizzle, all in one!
....this, as you rightly say
I remember watching a documentary on the death of Senna and they talked about the down side of using ground effect, while it is pretty effective and means you could use smaller wings reducing the effect of following a car, it rely's on the cars being a precise distance from the ground and any upset in ride height would mean going from 100% down force to just the wing down force in milliseconds...pretty scary (see dangerous) on a long high down force corner like the Parabolica for instance
Active ride would help there as well. Though I would take issue with your example of Parabolica at Monza - the initial turn in is where you need serious downforce and it's an early one there, once your speed is increasing (and with it the downforce generated by GE) you're pretty much winding the steering off and pushing the Warp Speed Now™ button for the main straight.
A corner like the old Tamburello is where you
really don't want to see a GE failure - high speed turn in, foot to the floor and very high lateral G loading. But since Tilke and the FIA between them have done away with virtually all of those corners....