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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.
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.