Europe Sans Car

None of those well known places you listed here require a car.
When were in the Provence area we went to the Tourist Info Center in Avignon and bought a escorted half a day trip to some accessible only by car towns like Gordes, Les Baux, Roussillion. St Remy

I am a bit confused by what you mean by "mother nature". Many of your destinations like London, Paris, Prague, Rome etc are big cities. They do have some nice parks, but these are still man made parks not wilderness.

But on the question of cars, I do not think you need a car for any of the places I am familiar with here. In big cities like London, Paris etc there are excellent public transport systems. In the Swiss Alps, trains can take you to many places including some that cars cannot go such as Jungfraujoch. Plus there are cable cars to many mountain tops which are also not reachable by car.

We have rented a car for driving through Provence -- picked up at Avignon TGV, based in St Remy and wandered the small towns.

Also Tuscany and Normandy/Loire Valley, but those are not on your schedule. Trains and planes between other places, and local transit (and walking) within the cities.

You have not given an itinerary. That is a list.

An itinerary is a day by day indication.

Day 1: London - changing guard, Tower
Day 2: London
Day 3: Chunnel to Paris

etc.

Lay out an itinerary, including what you wish to accomplish at each stop

I've been to all the places you list and have never rented a car in Europe. You will have no difficulty at all in most of those places. In a few (including Provence and parts of mainland Greece) you may find it awkward or impossible to directly link one very small town with another, which can require extra time, use of private transportation (like a taxi) or removal of a spot from the tentative itinerary. However, you do not mention any small towns in your original post, so it appears you may not plan to go to any problematic places.

The devil is in the details, so you really need to dig into guidebooks and be more specific about what ground you want to cover. All non-rural parts of Europe are accessible without a car, and you can get around even most of the rural parts to some degree, even quite small towns, by train and bus if you're selective in which small towns you choose. It would be more efficient to ask which parts of Europe are challenging or impossible without a car than to ask whether you've missed any that are perfectly fine to visit via public transportation.

Additional research should be done before you buy your flights, because I'm pretty sure you're going to discover lots more great stops in those areas (Seville, Cordoba, Granada and Toledo in Spain, just as one example), which will push the full list to more than 90 days in the Schengen Zone. I suspect you've got two trips there rather than one.

Places you haven't mentioned where I did feel constrained without a car included the Dordogne in France (though I still got to all but a few microscopic villages I would have liked to see, and that was partly due to limited time rather than total lack of transport), western Scotland (where you can't respond to sub-optimal weather very well without a car), and--for a future trip--the fjord country in Norway (where I can see I'll be spending extra time since I'm depending on public transportation--and that's costly in Norway because lodging is very expensive).

Ditto acraven!
I'd end up splitting that list into two trips by the time I delved into specifics. I'd also eliminate those that complicate travel (Prague may be an outlier, for example).
Parts of Greece require no car, but it totally depends on where and for how long.

Of the areas you have mentioned, to me the one that is best viewed with one’s own vehicle is Provence, but I have also heard of people basing in Avignon or somewhere and taking day trip tours with your companies that can get you out into the beautiful countryside.

Lucerne Switzerland is lovely and you can get a lot out of it without a car - boats on the lake, lifts to the mountains, trains or buses to other towns.

Provence can be done without a car depending on what/where you want to see. The country side and small villages are difficult without a car, but the other poster's suggestion for basing in a city like Avignon and taking organized day trips can work well.

Greece, outside of Athens, is a bit difficult without a car as there isn't that much train service, but there are buses to get to other towns and sites.

To see the nature areas outside the city you will most likely need organized tours. Within and near the cities, especially cities like Paris, London, and Berlin you can see a lot of nature using public transportation.

Am I missing any places that are perfectly fine to visit without
having a car?

Yes, it would probably be easier to list places in Europe that are not perfectly fine to visit without having a car

We've done 19 trips to Europe for a total of 54 weeks and have rented a car just once, for four days in Tuscany. The only caveat is that you need to allow more travel time when using public transportation. I would heartily recommend the Italian Dolomites for a nature fix.

If a traveler is covering substantial distances between major cities, a train is likely to be faster than driving. Sometimes it's much, much faster. That would be the case for Paris-Barcelona-Madrid, for example. And that's before you start allowing for stops along the way, looking for a parking place, and time at car-rental agencies. It depends on whether you're thinking about a trip covering major cities or bouncing around a more limited area, stopping at smaller towns and cities not served by fast trains.