Automatically share the project with the user who submitted the original request at the time the request is converted. This would allow various users to get access to all their requests without manually checking this box each time.
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It sounds like this might be a feature request and less of a question for the community. If that's true, you should totally submit it as an idea in the Innovation Lab so people can vote on it. Then come back and update your thread with the link if you want to give it a little promotion.
I think this is an option in the Setup area under Tasks & Issues, unless I'm misunderstanding what you're trying to do?
Primary Contact is typically the submitter of the request, though I suppose it is possible to designate someone else if your request queues are set-up that way.
It's also possible to configure it to behave that way for only certain Groups, so it would be possible to be creatively selective about when it would occur if you didn't want it to be a global setting.