Yes, the monarchy is the least powerful. We live in a constitutional monarchy.
I read that Guardian article some time ago. I also saw a legal opinion on it.
Some clarification:
Yes, the Queen may well have lobbied for changes. Many businesses organisations and other interested groups do.
Queen's (now King's) Consent (and Prince's, where relevant) applies to bills affecting the royal prerogative, and the hereditary revenues, personal property or personal interests of the Crown. This consent is not something that can be refused; it is part of the procedure for a relevant bill passing through the Houses of Parliament, and now happens at the third reading in each house.
If there was ever a threat to withhold consent, that would be a constitutional crisis.
King's Consent
Securing exemptions: I would need to reread the whole article to refresh myself on exactly what the Guardian was saying - from memory it was inaccurate.
It's important to remember that the longstanding principle in the UK is that the Crown is exempt from statute ('the Crown is not bound by statute save by express words or necessary implication'), so exemptions would not be necessary.
Of note, since 2010 this has not been the case for Acts of the Scottish Parliament, and since 2020 for Acts of the Senedd.