I want to get lost in France for at least three weeks

Yes, this is possible, but I guess only if you have your own vehicle.

I have done two road trips together with 3 friends through France experiencing all the things listed, except for the vultures. On both road trips we didn't have a clear goal in mind, and chose a destination from day to day. The first road trip was around the Vosges mountains, on the second road trip we went south from the Netherlands without a clear city in mind. We ended up in Montpellier. Both road trips lasted for about 10 days, but I could easily do it for three weeks.

At the end of every day we escaped the main high ways and started driving on the smaller roads in France to find a proper wild camping place. We used the map to look for green areas with a river or a lake next to it. At that time I didn't know but wild camping is mostly legal in France

Almost always did we find a proper place (if not an official camping was our plan B), but it did cost us quite some time. Some suitable places are fenced off, other places don't have a road nearby.
Sometimes we could park the car next to the road, walk through a thin layer of bushes and trees and find a huge field or a small lake very suitable to camp by and relax.

Because we didn't plan and didn't pay much attention to the map I can not name the places we stayed close to, but I guess, as I tried it from north to south, this is possible through all of France.

TL;DR:
Yes it's possible if you have your own car, stay of the main roads and look for green areas on the map.

Don't forget that you are not alone to look for quiet places. Everywhere that you can go to with a car, other people can as well.

Therefore, I would recommend remote places that are not accessible by car, or at least with a restricted access. For example:

• remote parts of mountainous regions such as the Pyrénées, but stay near the French/Spanish border. The path is called the HRP, Haute Randonnée Pyrénéenne. Moreover, you'll get the vulture experience.

• rivers: you can avoid crowds and listen to the silence by descending the Loire river with a canoe. Choose a period with calm waters and no holidays.

I'm an American living in France and have a summer home in Auvergne (Parc Naturel du Livradois Forez), and family living in the Ardeche (in the mountains, not the canyon). The Massif Central truly is deserted in many places, so I'm not sure where you went with floods of tourists, although of course farmers live there so you won't be sure to be completely alone all the time. Try wooded areas or mountains. Follow the advice about leaving the main roads. In Auvergne, try areas around (but not in) Ambert, La Chaise Dieu, and further south towards Le Puy. Ask in the tourist offices of the 'Parcs Naturels'; they can be very helpful and these parks are set up to preserve wild areas. In the Ardeche region, try areas above Vals les Bains in the mountains. Go to a small town such as Burzet and then make your way upwards.

Yes, there are tons of places!

One example: If you enjoy hiking, you could do a 10-day tour on the R. L. Stevenson Trail, which takes you from Le-Puy-en-Velay to Ales in the South of France, through the Cevennes National Park (officially the least populated area in all of France).

See detailed description on Walking in France.