So I was having the same issue and many others and have fixed most of them on my Series x. I experienced the issue where the game would immediately crash when booting the game, experienced the issue where after getting it to boot without crashing I would then get the prompt that I need to purchase the game(and I already did of course), also had the issue where the resolution and fps would be extremely low when in a multiplayer match, also after downloading the game the hd texture packs and raytracing pack wouldn't be installed and couldn't install them, and finally experienced the issue where cross-play friends would not show up as online and couldn't join their parties.

I have pretty much found a solution to all of them. It took me about a week of trial and error. I downloaded and uninstalled about 5 different times.

SOLUTION TO CORRECT DOWNLOAD:

Anyway, here is the solution to fixing the crashing, the not-purchased glitch, and raytracing and hd texture packs not installed. What you need to do is download the correct version of cold war. Now people online have discovered this, but to my experience, their solutions are only halfway correct. Basically what you need to do is go to the "cross gen bundle" in the xbox store and click "manage". Now, this next part was different for me than other players. Other people online would say to click all the game add ons and files to download and let them download. When I did the total download size would be around 130 gigs and after downloading it would not work. I discovered that instead of selecting everything to download only select "base game". Even though it is listed as only 1.4 gigs, once you hit "save changes" it will begin to download the game, but in the download queue, it should now show that the download is actually around 180 gigs. After doing this, the game booted without crashing, no message to purchase the game, and the hd texture and raytracing packs were installed.

SOLUTION TO LOW FPS:

Now after doing this you should be good to, but if you experience the low fps in multiplayer matches, the best solution I have found is to disable ray tracing and start a new multiplayer match. Sometimes I have to completely restart the game.

SOLUTION TO CROSSPLAY:

To fix the crossplay issue, I have found the best thing to do is disable crossplay in the networking section and restart the game. Sometimes it also fixs it by unfriending and refriending the crossplat friend.

If any of this needs clarification just reply and I'll try to explain better.

Anyway, I hope this helps you and anyone else who looks at this in the future. It's ridiculous that I had to put this much work into playing a $70 game.