We visited Mongolia this summer and I did a lot of research into "luxury" travel. We decided not to do the Eagle festival or Naadam festival and wanted a visit with less crowds. I don't have personal first hand experience with these tour operators but my research found the following US based tour operators who were responsive to my inquiries and had good websites: Nomadic Expeditions (reputedly the founder of tourism in Mongolia and started the Eagle Festival), Geographic Expeditions, and Wilderness Travel. National Geographic also has tours to Mongolia.
We ended up using Nomadic Journeys, based in Ulaanbaatar, which also has tours to the Altai Eagle Festival which was excellent and can arrange for luxury accommodations, and which we found to be cheaper than the US based operators. In our travels we met Mongolian based guides and their visitor clients from Mongolian Luxury Travels and Amicus Travels (while stranded at the airport due to cancelled flights) and the guides seemed to be competent and their clients satisfied with their tours. Google searches can be informative but there isn't a lot of information available about luxury travel in Mongolia because the tourism industry appears to cater to the travel who wants to trek, diy road trips, and camp/backpack. See this interesting recent article: https://www.nytimes.com/2023/09/13/travel/mongolia-millennials-road-trip.html.
We wanted a private guide and driver and best available accommodations with ensuite bathrooms. Since the tourism infrastructure is still developing, travel in Mongolia is a challenge, but its a wonderful country and the people are friendly.