Not really because the Scotish parliament doesn’t need an election to change the government. The whole Scottish system is designed that no one party should get a majority, at the time because they feared a permanent Labour government. Except the SNP broke the system just the once. In general though, and currently,Scottish MPs are always in a position to change the first minister if they choose to, without an election being called, unlike Westminster.
If Holyrood doesn’t want the new leader to be FM, then they won’t be.