Does "early boarding" imply "you get a space in the overheads"?

Does "early boarding" imply "you get a space in the overheads"? Many airlines do offer "early boarding" as a paid add-on or as a perk of status or booking class. Many people equate this with "I don't have to fight for overhead space", i.e. you are guaranteed to use your carry allowance (provided it's within size and weight limits, of course). I recently flew Lufthansa domestically and was in Boarding Group 2 (Star Alliance Gold). At the gate an agent was walking through the crowd and randomly telling people to gate-check their carry on (full flight). That included me. I argued that Group 2 should pre-empt me from gate checking but she wasn't having it and I didn't want to argue too much. To be clear: My carry on was regulation size and weight. I was in full compliance of all LH rules about what you can bring on board. When asking the agent why she picked me she wasn't able to articulate anything meaningful. Question: Can I insist on bringing compliant carry-on on board if I have early boarding privileges? If airline matters, LH is preferred but other airlines are welcome too.