If the ONLY think the Illuminati did was stay hidden, Steve probably wouldn't have a huge problem with them. The problem is that they did a lot of questionable things behind everyone back. That was the REAL issue. I think the difference in our opinion lies in you only looking at that single aspect of them staying secret and not at the other things they've done. Hding from everyone is frankly the tip of the ice berg. Starting a war with the skrulls which led to Secret Invasion, exiling Hulk into space which led to World War Hulk, and brigning the Infinity gems to earth which allowed Hood to collect them was the bigger issue.
But yes, Steve did decide to keep them guarding the gems a secret. Stark faked the gems destruction, so people would think they were gone. That meant that they couldn't advertise the fact that they had the gems. They kept it a secret, something Steve apparently agreed with.
It's not until they decide to blow up planets that Steve ends up on the outs with them again. Steve is tactically flexible... just not morally flexible. He'll adopt another persons strategy if it make sense to do so... but not to the degree that he would compromise his ethics. That's where he becomes "stubborn."