Sounds dangerous. No, definitely not normal. My '92 300E is dead solid in heavy rain.

Sure at that mileage it could be suspension related. Things do wear out. Have ball joints, tie rods, steering shock, and rear suspension links checked by someone that knows what they are doing.

I had an issue on my former '90 300E that was due to the car not being aligned properly. Dangerous handling in sweeping curves.

Have an alignment done at the Dealer, not at a generalized alignment place.
They need to have and use a spreader bar to get is right.

You said three sets of tires. Tires is a pretty broad term. Hopefully, they are a good brand in the correct size and rating (at least H) and not cheapo's. Cheapo's could do that.

Steve