I write this post more than one month later, but the anger still is present.
They say that in Yellowstone come more than 4 millions people, that’s way booking well in advance is mandatory. I fully understand.

But 99.99% of these people drive in the park and change on parking site with another.

We tried to book on internet, but it is not easy at all to find information about the trails. As we try to be as free as possible, we decided to go directly to the ranger office at the entrance in the park. We have to book all the camping sites on our trail. But the information on trails is only the millage, not the elevation, or time needed to get from A to B. And there was no description of the tracks. The ranger (volunteer) recommended us to go to a book shop to search in a certain book more information. I was thinking to buy that book and donate it, but it was too expensive for such a gesture. We select a couple of trails.

We come back to the office and there the ranger starts to call to check that each camping site on our route is free, the IT system was down for a couple of days. The majority of sites were booked. The camping sites near the lakes are booked by fishermen. So we had to change the route and the sleeping points due to those restrictions.

We are happy to finally go. But how can I explain that during 5 days of backcountry ( in the mountain, not on the highway), we met maximum 3 groups of hikers ?
Everyone told us the same story, that they camped alone on a site with 6-8 places.

there are nearly 300 back country camps, so if there is only one group in each, lets say 4 people, there are 1200 hikers per day. If on average, a hiker stays 5 nights, and there are 5 months of good weather, that means there are about 36.000 hikers / year, so 0.9%

why do they make so difficult for the hikers to plan their trip, and to be free to chose, when the resources are so poor exploited ?

they say to stay on the trail, but quite often it is very poor marked.

Sometimes we walked like in a desert, other time like in a swamp, with high grass. Once we could not find our camp site, we passed a group and they said they couldn’t find it too. Another time we spent 20 minutes in a heavy rain trying to find the trail, luckly I got GPS signal.