So from reading through this whole trail this is how I figured out how to make the change to the project lead on Jira (thought I would put it all together in one Answer).
1 - Open the Project and go to Project Settings (Screenshot 1 below )
2 - This will open up a 'Details' screen you can scroll down to 'Project Lead' field located under the Description section. You can then type in the name of the new Project Lead (Screenshot 2 and 3 below)
3 - Once name is updated you just 'Save Details' on the bottom of the screen.
Then if you look at your Project Settings>Summary screen you will see that your Project lead has been updated. To note I do have Jira Administrator permission (though I don't know if this is required do update the project lead field).
This is a known issue for Next-Gen projects.
If you want the simplest solution now, please try our free app Classic Settings.
The app will let you set the Project Lead exactly like Jira does in classic projects. Find it under: Project Settings > Apps > Classic Settings. We also support Default Assignee, and more feature are planned to be released this sprint :)
Other than that, you can vote and keep track of this issue. More complicated solutions using the API are documented there in case you prefer to avoid the app, but I'm hoping you'll give us a chance. It's fun to have the classic experience after the long search.
Go to the "Administration" page for the Project. Then in the "Roles" section of the page, click on the "View Project Roles" link. If you have the correct perms, you should be able to change the Project Lead by clicking on the small pencil icon next to the name of the current Project Lead.
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