Much of my European travel has been done for business; kind of like a medical device traveling salesman. So, I can tell you that not only is it possible to do this but a great way to really see a country.
Some thoughts from experience:
Cars are a liability in large European cities. Traffic and parking suck. I would typically park the car and cab it around to clients as a way to keep blood pressure lower. City driving can get pretty aggressive. Spain and Italy being the craziest.
Good way to check out neat cars if you are there long enough. Drive to a city, turn car in and get new car for next city. Most cities have a transit pass for tourist that is a bargain.
In My Opinion, the best of Europe is to be found outside of the large cities, so this is where the car really pays off.
Rent a small car that gets good mileage. I have seen how US citizens cry over high gas prices and let me warn you that European prices are nuts in comparison. Besides, "Slow cars fast" and all that is true. I had more fun driving a 1200cc rental on the Autobahn that I would have the Mercs and Ferraris that were flying past me. When was the last time you literally held the gas pedal to the floormat for like half an hour looking for another 2 mph!
Inspect your car for damage before you leave the lot. My experience is that European renters are more aggressive at charging for wear and tear damage than in the USA. You want to be insured for stuff cause door dings happen and Paris in particular street parking is a contact sport.
I never had anything other than my regular Canadian drivers license and it was never a problem.
Driving on the wrong side of the road is not as horrible as we fear. The scary bit is that most rental lots are going to dump you into busy traffic right away like an airport or downtown train station. So yeah, bit of pucker factor for a while.
Adopt a zero alcohol policy while driving in Europe. It is just a good idea.
Speed cameras are a thing. Your GPS will try to warn you and yet surrounding traffic is still going 100 miles and hour, so yeah. The ticket goes to the Car Rental place and they tack on their own fee that ends up on your CC as per small print. Funny story, rumor was that Italy had a toll road that timed your transit and handed out fines at the toll exit. Supposedly there would be Ferraris pulled over for picnics on the side of the road waiting for the time to reach legal speeds.
Leave nothing in the car overnight if you want to own it in the morning. Rentals are easy to spot and a target for those that do. Another reason to rent lower-end small cars.
So yeah... Go for it.