Wow, I actually asked the very same question on Reddit some times ago and got some truly enlightening answers. So I think I'm well qualified to discuss this.
As the contributors above have said, the Stewards of Gondor were bound from claiming kingship by both tradition and the security of their position.
"Each new Steward indeed took office with the oath 'to hold rod and
rule in the name of the king, until he shall return'. But these soon
became words of ritual little heeded, for the Stewards exercised all
the power of the kings. Yet many in Gondor still believed that a king
would indeed return in some time to come; and some remembered the
ancient line of the North, which it was rumoured still lived on in the
shadows. But against such thoughts the Ruling Stewards hardened their
Nonetheless the Stewards never sat on the ancient throne; and they
wore no crown, and held no sceptre. They bore a white rod only as the
token of their office; and their banner was white without charge; but
the royal banner had been sable, upon which was displayed a white tree
in blossom beneath seven stars."
Tolkien asserted that Gondorians simply will not accept a person (even highborn) as King of Gondor if he's not from the House of Elendil.
Now we've set the ground rules: Not from House of Elendil = No claim to Kingship of Gondor. But was any person from the House of Elendil eligible for kingship?
"On the death of Ondoher and his sons, Arvedui of the North-kingdom
claimed the crown of Gondor, as the direct descendant of Isildur, and
as the husband of Fíriel, only surviving child of Ondoher. The claim
was rejected. In this Pelendur, the Steward of King Ondoher, played
the chief part. 'The Council of Gondor answered: "The crown and
royalty of Gondor belongs solely to the heirs of Meneldil, son of
Anárion, to whom Isildur relinquished this realm. In Gondor this
heritage is reckoned through the sons only; and we have not heard that
the law is otherwise in Arnor." 'To this Arvedui replied: "Elendil had
two sons, of whom Isildur was the elder and the heir of his father. We
have heard that the name of Elendil stands to this day at the head of
the line of the Kings of Gondor, since he was accounted the high king
of all the lands of the Dúnedain. While Elendil still lived, the
conjoint rule in the South was committed to his sons; but when Elendil
fell, Isildur departed to take up the high kingship of his father, and
committed the rule in the South in like manner to the son of his
brother. He did not relinquish his royalty in Gondor, nor intend that
the realm of Elendil should be divided for ever. '"Moreover, in
Númenor of old the sceptre descended to the eldest child of the king,
whether man or woman. It is true that the law has not been observed in
the lands of exile ever troubled by war; but such was the law of our
people, to which we now refer, seeing that the sons of Ondoher died
childless." To this Gondor made no answer. RotK, Appendix A, I, iv,
Gondor and the Heirs of Anarion
The above passage shows that the Stewards of Gondor might not be able to be the King, but they can determine the eligibility of a potential King. And they denied Arvedui of what's supposed to be his birthright.
So in a sense, this does confirm your opinion 'men in that position enjoyed the power.' The Stewards certainly enjoyed being essentially ruler of Gondor. And Denethor II was fully prepared to deny Aragorn's claim.
Security of their position:
The position of Stewards was a position given by the King of Gondor. And that was the ultimate legitimacy of Stewards. They can sit in their chair at the bottom of the High Throne and rule Gondor and none shall question them because the King of Gondor commanded the Stewards to rule in his absence. And the convenient thing about how there hadn't been a King in Gondor for 976 years? It does nothing to disprove the Stewards legitimacy!
Now if a Steward actually declared himself 'King of Gondor', he shall no longer be legitimate -- since the last King of Gondor didn't add 'candidate for the Throne' to the Stewards' job description. The lack of legitimacy would made him vulnerable to political attacks and revolts.
So why move yourself from a perfectly legitimate position to a not-so-legitimate position when you have the same amount of power either way? Why expose yourself to such risks?
Connection with this Earth:
Since Tolkien's actually a human (not Gondorian nor Sindarian nor Dwarven), he actually based his writings on some human societies of this Earth (then idealized them).
Roman Republic, renowned for its distaste of kingship, lasted half a millennium solely on the ideal of democracy -- a kind of reverse parallel to Gondor's empty throne.
Japan, during its Feudal Era, went through a period when its Emperors were basically puppets sitting pretty on a throne and the shoguns controlled the country. But the shoguns themselves never replaced the Emperors, because the Emperors are the 'descendant of the solar goddess Amaterasu' and that bloodline is the only one that may claim the title of Emperor of Japan.
While there might be many cultures that worshipped strength over blood (China's ever changing dynasties being a prime example), we assume Tolkien's Gondor had a culture similar to the ones mentioned above.
And I hope that answers your question!