10-15 years is a lot to ask no matter what budget.
Don’t get me wrong it may still work completely fine, i have used older parts, but if you plan to play new games you’re gonna end up struggling. 10 years ago the best gpu was a gtx 980 and its barely holding on right now.
If you plan on playing a very specific set of existing games and don't play new fancy titles, you might pull it off though. But thats not the vibe i am getting here.
With that budget you can straight up get a rtx5090 (if available anywhere) which is currently the top tier consumers gpu on the market but chances you are still going to want to upgrade over a year of 5 but its definitely gonna last longer then any other (in theory).
Which bring me to my best advice, the main advantage of your desktop isn’t just performance it is upgradability. Providing a list of software and games you use you can easily get a pc perfect for that, saving some money for the future when those needs change. Then upgrade the part that needs it. Bonus is eventually your old parts will eventually end up becoming a secondary pc.
Which bring me to my last point, if you are serious about streaming you may want to split that budget into 2 separate systems. One to play the games on, and another tailored to streaming and twitch. Many streamers do it this way for practicality and performance.