In the absence of a king, the stewards had the same power and authority as the kings and the position was hereditary. The stewards were effectively just a dynasty in the kingdom. Aragorn's ascension to the throne didn't really impact the way Gondor functioned.
The real republic is actually The Shire, which is consistently depicted as the best place on Middle-earth and the only place that regularly produces people capable of resisting the One Ring.
Fair enough, but that existence is allowed because the Rangers of the North, last reserve of the blood of Numenor and the Kingdom of Arnor, shelter it.
It also gets absorbed as a territory of Gondor/Arnor when man briefly reaches the heights it used to have before declining again.
I think you might argue that this is Tolkien expressing his concept of whatever the hell "anarcho-monarchism" is supposed to be, which is apparently the chosen militants of God sheltering pastoral British villages from reality without even taxing them for the privilege.
I'd also like to introduce you to the innately British concept of the "monarchical republic":
You'll note that Elizabeth I is listed as a potential example of this idea, though I tend to agree the concept doesn't match the historical reality and it would take becoming a constitutional monarchy to resolve the inherently oxymoronic nature of the systems.