Under the terms of the Oath, yes. She swore to preserve their rights and privileges. That is one of their rights and privileges. It's not implied, it's explicit.
Royal prerogative still exists; only in the case of abolition of same would the monarch be duty-bound to rubber stamp laws that lead to the detriment of the kingdom.
Sorry, but you clearly don't know as much about this as you think you do. The Monarch has the power to refuse Royal Assent only on constitutional issues. They do not have that power on religious issues - in the 1820s (I don't know the exact year off the top of my head) George IV wanted to withhold Royal Assent from a Catholic Emancipation Bill precisely because he believed it violated his Coronation Oath to uphold the Supremacy of the Church of England. It was made clear that he could not do so - that once the law had passed Parliament, he had to act solely on the advice of his Ministers.
The last time a King even considered withholding Royal Assent was nearly a century ago - and that was over a purely constitutional matter (the Government of Ireland Bill of 1914) - and he determined that even on constitutional matters, where the power does still exist, the only situation it would be justified to withhold Royal Assent is to prevent a national disaster.
As I said in a previous post, it's amazing me the number of American Republicans (and I don't know if you are one or not, but it is certainly a large part of the American conservative grouping) seem to think the British Monarch should act as a tyrant and overrule the elected Parliament of the United Kingdom at the moment.
The way I read it, the rights and privileges remain intact only as far as it allows the monarch to conduct the duties of “maintain(ing) the Laws of God and the true profession of the Gospel” et cetera; and this absolutely gives the monarch the right to exercise prerogative and reserve power on religious issues. Implying that the clergy or the parliament holds its powers in perpetuity, with absolute right of abusing them (implying a totalitarian bent on their part), is implying an abolition of prerogative where none exists.
Remember, this oath is sworn to God.
Equally amazing is the number, in this and the other current threads on this subject, who are apparently ignorant of the distinction between assent and approval: and that the one does not imply the other - in this or indeed any other context. The sloppy journalism of the various press articles quoted, which have assumed 'approval' where none exists, does nothing to remedy that ignorance.