France and the United States are the notable exceptions where the president is more important than the prime minister.
I don't know of any cases among the developed English-speaking countries with Westminster-style governments where your latter sentence is meaningful. The Canadian Prime Minister, especially, is said to be the most powerful single government leader in the world, but really is merely the (to coin a phrase) first among equals among his peers in other countries.
For others' benefit, a Canadian Prime Minister can
* Appoint anyone he wants to cabinet positions
* Appoint anyone he wants to the Supreme Court[1]
* Appoint anyone he wants to ambassadorships and other high positions
* Sign any treaty he wants
* Call an election whenever he thinks he has the best chance of winning more seats than his party possesses in Parliament at the moment
* Get any law passed he wants (assuming that his party has a majority or equivalent thereof), with no meaningful need to deal with an upper house
* Run for reelection as Prime Minister as often as he wants
Most of the above apply to the UK, Australia, New Zealand, and Ireland. The Irish and British upper houses have little power, and New Zealand no longer has an upper house; only Australia has an upper house in the legislature with significant power.
[1] Yes, yes, I know about the convention that the PM names someone from a list of suggestions. That's all that is; a convention. The only actual requirement is that three of the nine justices have to be from Quebec.
One advantage is that you know who to blame, especially when it elections come around. Instead of the "dead lock" that can occur when both the Legislature and Executive say they have 'mandates' that are opposing to each other.
I don't think you can be a Cabinet member and a member of Congress. As Cabinet members are officers of the United States and you cannot hold another office while also being a member of Congress (Article 1, Section 6).
It's also why the Presidential Succession Act may be unconstitutional. As the Constitution specifies that only an officer may be designated a successor and members of Congress are explicitly excluded from being officers.
Given that the first time the Speaker of the House and the Senate president pro tem were proposed as successors under Article II was during the literal 1st Congress in 1792, I think it’s pretty unlikely that any modern court would find that to be unconstitutional.
“Many of the actual people who signed the Constitution thought this was okay and demonstrated their belief by introducing legislation to that effect” is about as ironclad a precedent as you can hope to find.
Even when the 1792 version of the bill was signed, the issue was brought up. By the "actual people who signed the Constitution".
And we've never had to actually test the Presidential Succession Act, which is where the courts would decide if it actually meets the Constitutional standard. The fact that it is a law, has been a law for a while, and a version of it has been a law since nearly the beginning of the country has no bearing on whether or not that law is legal.
This is true! But it’s also immaterial to the point I’m trying to make.
I’m not arguing for or against the de jure constitutionality of the law; I’m saying that it’s unlikely any modern-day court would find it within themselves to overrule the de facto state of affairs that was created by the framers themselves.