IIRC the Queen only exercised the prerogative to prevent a bill effecting the Royal Prerogative from being considered on the advice of the Prime Minister; removing power from the Crown has the practical effect of removing it from the government of the day.
IIRC, there was an episode way back when (70’s?) when the two houses of the Australian government controlled by different parties were deadlocked. (Good thing that doesn’t happen in a republic like the USA? 
) The governor general exercised his power to dissolve parliament and call new elections when the deadlock could not be resolved. Usually, as a reticent representative of the Queen, the GG only does that on the “advice” of the prime minister.
If I were to imagine a conversion to “Republic” for a country like Canada, or it’s provinces, the Queen and her representative GG would be replaced by an elected “president”. Then we’d be in the same boat as has happened Israel or Italy, where the moderately unpowerful president be a recycled tired old politician and would likely be facing corruption charges. Plus - the upper house of Canada (Senate) is appointed. The argument against it being elected is that a competing house claiming a mandate from the people would likely exercise more power and possibly deadlock the government, as happened in Australia. The same might be true of a president - an elected personage with the power to veto bills and/or call elections might claim that their mandate derives from the masses, not some farcical aquatic ceremony, and proceed to exercise that power. The Queen is like Daffy Duck - she has a very neat trick, but she can only do it once. Then she gets replaced; any governor general who tries that trick will likely be quickly replaced by the Queen on advice of the country’s prime minister.
As for Cuba - why not? My point was that the pretense of democracy and the tricks used to ensure that the “right” person gets elected, would probably not hold up once that state were subject to the constitutional guarantees of the USA and its power to enforce those laws.
The Vatican isn’t realy a Republic since the citizens themselves don’t vote for their leader. I don’t see why a governor-for-life could not be a choice for a state; but once again, the trick would be to defend that system against the will of the people should they want to change it. What’s the criteria for calling a constitutional convention or modifying the constitution? The fun fact about the parliamentary system is that unless there is a constitution, the parliament can do anything it chooses. Plus, assorted levers were used in Britain to reduce the powers of the monarch and the House of Lords. Presumably, it would not be difficult to do so in such a state, using the financial levers available to the legislature. I presume, ultimately if there were an insurrection against the state government the feds could step in and impose a new constitution.
Only if you equate ‘head of state’ with ‘executive power’ (or ‘administrative function’).
In the USA, the ‘head of state’ function is also held by the ‘head of the executive’. In some countries they are both directly elected: in other countries one or both is directly elected: in other countries something else.
You then get into the argument about if the symbolic power of the Queen is executive, and if the reserve powers of the crown are executive. At which point the argument becomes something like the argument over when you should have ‘end of century’ celebrations (at the end of 1999? Or at the end of 2000?)
I guess that in a way you are right - nobody has truly defined what the constitution means, because it has never been necessary. Therefore, they have never explicitly defined “Republic”. However, given what the framers had been through and fought against, and taking into account the prohibition against titles, etc. - the framers were explicitly forbidding monarchy and/or nobility as the executive, which was the competing executive arrangement of the times - so a republic is the opposite of a monarchy. Then we can get into the debate whether “President for Life” is a title vs. an office…