Yes, your are right in some cases. But just to show you that there are indie fps shooter games also available:
http://www.alpha-prime.com/

This had a much lower budget and I personally enjoyed it very much. They did not use any of the big engines and they licensed some middleware for tasks like AI. But it is a very good game with lots of additional gameplay options like controlling vehicles, cameras, remote control of many objects, indoor and outdoor scenery and some modern visuals especially at the time of release.

So I am sure it is doable. Yes, it needs some very ambitious people, time and at least a decent budget. But it can work even without the budgets you mentioned.