This seems a
staggeringly vaguely bad idea.
I'll admit I don't know what your "taylorism" is, but let's outline why I dislike it.
Significant detail has been added to the question - including having multiple groups, and the limiting terms & no re-election of past winners.
Even with this, and these are welcome changes, I still have reservations about the election process. The staff in your team don't know how popular they are - some will think they're more popular than they are, for example. I see no benefit for these people to discover the truth - it is not going to get them to change their ways, only feel worse. Those who are more popular than they thought won't work harder, and worse, what if the most popular person isn't the hardest worker?
What is that going to do for the culture of your team - this is going to have an impact, which you are unaware of, and the impact is utterly out of your hands. Why on earth would you want to gamble like this?
You should just let all team members know the plan, know when they're scheduled to manage a team, inform them the schedule can change, but is generally a good guideline (and not change the schedule unless there's a good reason).
You could consider having three teams after the first three months - not only will this get you through the entire team faster, but you can let the previous two leaders manage the leader of the third team. This would create a lengthened training process (people do learn when they train!) as well.
It's because of the election process. Everything else seems sound.
You're effectively making your workplace into a popularity contest. Everything else you suggest - the 3 month periods, the mentoring, seems fine, although with 14 staff will take you several years to give everyone a shot - might you not break the 14 into two groups of 7 apiece, each with a leader? It's unwieldy for a manager to have more than ~5 direct reports anyway. This way you can get through everyone in less than 2 years. Breaking into groups of 5 will yield even swifter throughput, although that depends on how much bandwidth you have for mentoring.
But, back to the election process - what is going to happen to the least popular members of your team? Exactly. They're
- never going to get a chance to develop into leaders
- going to be aware they're not getting the chance
- going to be aware they're not getting a chance as they're not popular
- going to leave, with a bitter taste in their mouth
You can break staff into 4 groups - along an x-plane of hard workers, and a y-plane of talent. You've got then high talent, hard working, high talent & lazy and so on. The mistake many managers make is focussing on the high-talent hard workers. Yes, you want to fire the low talent lazy people, but the low talent hard workers and the high talent lazy people make up a large chunk of your organisation. Why not train them? Worse, giving benefits to the A-tier types is often wasted - they're the people who are going to to and get extra training on their own anyway!
Your plan of popularity falls into this trap - you're not going to be focusing your management efforts on training those who would most benefit, but probably the star performers. So it's an interesting combination of disincentivizing a large bloc, and ignoring another large bloc.
So almost all cons from me, I'm afraid.