Is there a special system for bypassing flags to get officers promoted?

Only way I can think of to have this happen is if there was some form of appeal or correction to military records that was approved, with an as of date of six days after the FLAG was imposed. The most logical thing would be a pending investigation FLAG that was never lifted. Maybe one for a referred OER that got favorably concluded. That's about it.
Common FLAGs like APFT or HT/WT fail would not be treated this way.

Chances are that the person was not on the same flag for three years. A three year flag is virtually unheard of unless you're awaiting court martial for fratricide. I can only think of two times I've seen someone flagged for three years or more and it was Nadal Hassan and Bozicevich. A three year flag would raise questions all the way at the HRC level.
As for the promotion, when someone is on the promotion list and becomes flagged, they receive a second flag removing them from the list. Once the first flag is resolved a board reviews the promotion and decides to grant it or not. The DOR will be the date it would have been granted had the SM not been flagged. Another possibility is that the date of the flag was changed and the day the SM would have been promoted was prior to the effective flag date. How or why that happened - I couldn't tell you without knowing more, but HRC has the power to do anything when the circumstances warrant and the chain of command supports.

If by “flags” you mean adverse information in an officer’s record, then I say “No.” some adverse information has a definite life span. It may be removed from the record after a period of time. Also the SM can appeal to the Service HQ for a change to record especially if the adverse information was entered due to discriminatory behavior by a more senior officer. What flags were you thinking about?