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I'm the weird guy who thinks that the runner up should still be vice president. Our system works by checks and bounces, so we should put pressure on elected officials to work with people they don't entirely agree with.
A threshold question is what role in government the Lieutenant Governor has.
If they're analogues to the Vice President, then their job is basically to be the President's replacement if something happens to them (and to save the President a walk down to Congress to break ties in the Senate).
In that role, I think it makes sense to let the Governor choose who will act if something happens to them, on the condition they're still elected as part of the ticket.
If the perform a separate role, then elect them separately, and the line of succession is incidental. >

I mean personally I'm against the independent executive on the general rule that it almost always attempts to seize power from the legislature without being able to be summarily dismissed like a legislative head of government usually is.
However having the possibility for the primary and secondary executives to be acting against one another is basically just asking for trouble, I think recently Idaho had a crisis where the Maganut lieutenant governor attempted basically a coup while the governor was on out of state business over said governor not being as big an idiot about covid protocols as she was. >

If the Lieutenant Governor is part of the state’s executive branch, and part of the governor’s team, yes they should. 
If the Lieutenant Governor is actually the leader of the state’s senate and really working as part of the legislative branch of the state, no they shouldn’t. 
That’s what should determine it in a state by state basis. >

No, I like this way, and think it could also be a good way of P and VP.
In fact, we might get better VPs instead of someone we'd dread if the P was gone. >

You will run into a similar problem that we saw with the original constitution issue with Presidents and Vice Presidents. The 12 amendment is there for a reason. Adams and Jefferson actively fought against each other while P and VP and it caused chaos. Similarly, LG still hold power and they need to work with Governors and Legislators in order for the Governor to be effective. >

I personally feel like having separate elections for Lieutenant Governors is better since voters are the ones deciding who will be the nominee through the primaries. But then again, elections for Lt governors vary by state. Some are elected on the same ticket with the governor, others have separate elections, and others are just given the title because they are presidents of state senates. >

Yes. The governor shouldn’t have to worry about what the LG is going to do if he/she has to temporarily run things.