No I do not think the Sheikah are behind the scenes in Hyrule. Mostly because they are essentially extinct. There are not enough of them to have much of an agenda beyond keeping themselves alive. However they do seem to have a position of high importance to the Hylian royalty. From Impa being a bodyguard of Zelda to Impaz being the keeper of the ancient sky book until the new messenger to the sky appears. She even says the villagers of the hidden village were a proud people that served the royal family.
This village was once the secret home of a proud tribe who served the royal family... But it fell into decline, and became infested with dangerous beasts. It's become an awful place..." — Impaz
The sheikah eye appearing in so many locations leads me to believe that, while they were not controlling Hyrule, they definitely had a part in the construction of many different secret areas. And were involved in a lot of the happenings all across Hyrule. But they were personally serving the royal family of that country. So it could be possible that all official projects were either led by or at least had a sheikah or two associated with.
The real confusion comes from so many different characters wearing the eye. Agahnim, Veran, and even Vaati in his monster form all wear the eye. But Agahnim's wearing of the eye might also be a part of his attempt to gain the king's trust. Since it would be familiar to the royal family as the tribe that serves them historically. And the same symbol is found in Hyrule palace at the time. As well as Zelda's cloak in TP.
Now what I have noticed is how the eye appears on individuals that are known to be skilled in magic. So the eye might have more of a magic symbolism than a tribe or alignment one. The Sheikah were known for their magical abilities and creation of many magical artifacts. Each of which has an eye inscribed on it. The Dark wizard Agahnim, the Sorceress of Shadows Veran, the Wind Mage Vaati, Madam Fanadi the fortune teller, and Princess Zelda, each is a magic user that wears the eye. And items that have the eye inscribed are normally magical themselves. Including stones found on the ground in Hyrule. So it might be possible that the eye is a type of symbol commonly used to magic use and magic users in Hyrule.