> For me it’s not so much about maintaining authority as it is about not letting the people who you’re trying to lead walk all over you anytime they disagree or push back.How not letting the people to walk over you is not maintaining authority? It is pure hierarchy game. "Who is more dominant male here?" Even if you are Buddha and can control your mind perfectly, separating lower instincts from rationality, then dude you are speaking to is not Buddha (most probably), and he would interpret situation as pure hierarchy game.
> How does just doing someone’s work for them indicate any kind of effective leadership?
When we see disobedient teammate we already know that leadership is ineffective. The only question remains: can the leadership become effective in the future.
> What lesson or mentoring does that provide?
It is generally good idea to reach several goals by one move. But not every time. If you cannot simultaneously reach work goals and education ones, then you can split goals and reach them in more then one move.
There is not much sense in trying to push while conflict is at its peak. Emotions get over mind on both sides. Possibly time is pressing and it is not the good idea to start discussion right now.
Later the lead can find time to speak about incident, with all its aftermaths, including him doing work for other (It gives the lead real power on conversations, like "Unprecedented! Due to your stubborness I has to do your job. Should I take your salary also?"). It should be time when the cause of conflict is in the past, when there are plenty of time to talk the past conflict through, when emotions not the present reality but tool in hands of the lead. And one more: when no one around except disobedient team member and the lead to exclude influence of social group effects on both.
At that moment the lead can calmly and purposefully explore his disobedient teammate motives and let teammate to understand the lead. Ether they will find mutual understanding, or not. In the latter case the lead would have all the reasons to fire employee who does not understand what team work is. In the former case conflict would be solved once and for ever, and there would be no need to repeat it again and again, which is real threat to authority of the team leader.
In contrast pushing conflict at its peak just to settle authority of the lead is not constructive. Disobedient teammate will learn nothing except his leader is a authoritarian headstrong dominant male, even if it is not so. The only consequence will be increased chance of successive conflicts with this teammate. It is simplier to fire him right now, than to wait for sequel.
Alternatively he will learn that his opinion is not needed, and will become silent obedient and non-creative worker. A kind of dump machine to type on keyboard. But if this is the goal of the lead, he'd better find another employee. Breaking human by force can be loud and unpleasant for the whole team.
There are no possible happy endings if the lead values authority above all and does not use his creativity to work with people.