This thread has gone a bit crazy but to bring it back around.
The directly elected Mayor model is just another form of executive power. In the Mayor model residents have the ability to directly elect the Mayor. This is opposed to the Executive model where the largest party is invited to elect a leader of the council. The Executive model will usually have a "council chairperson" or "Lord Mayor" who will chair the proceedings of the council. In a Mayor model the Mayor is expected to do that. In the Executive model the cabinet of the council is made up of councillors usually from the ruling party.
As a general fact check at the time of the referendum on the Mayor the Labour Group campaigned abolishing the Mayor knowing that it was the largest group (by some margin). That is no longer the case but seems to still be their position.
Topline pros and cons:
Pro-Mayor - directly elected so people can vote them out directly. Power of budget setting (but requires ratification by full council), can appoint a mixed cabinet, can be a different political party that the majority party which forces parties to work together, they are elected by the whole authority not just one ward.
Cons-Mayor - lots of direct power, leadership can't be changed throughout term, cross party working doesn't always happen, can become personality-driven, tends to be more costly, can just be another partisan face that brings no skill to the role
Pros-Exec - Leaders can be changed by political parties if performance is bad, likely to be more stable as party majority tends to be more stable year on year,
Cons-Exec - Only one ward theoretically elects the leader of the council, much less accountability, creates partisanship, lots more "special allowances" which tends to negate the cost of a mayoral system, no clearly elected leader role, creates a partisan "jobs for the boys" vibe (although the mayor can do that too), quality of Cllrs in local govt can be very poor
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That very briefly covers that. I personally don't have that much of a preference. There are some places where it works really well and the area has benefited big time from having a mayor at council level: Bedford and Watford spring to mind. Others not so good...
For a party like Labour (and many tories) it will always prefer the Exec model because its easier to win. I am not saying that this is why you are in favour of abolishing the Mayor but turkeys don't tend to vote for Christmas.
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On the abstention of the budget stuff - Labour could (and should have with a group of 18) come up with an alternative budget. Unless you thought the vote would not be as close. You could have used the voting block quite effectively to have the changes you wanted to see added to the budget rather than abstaining. But I don't know the specifics behind the logic here and I don't live in Boro anymore.
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On the purchase of the Crown - this would require a capital budget rather than the revenue budget so you can call the decision in and have it scrutinised (revenue budget being year-on-year spending commitments, capital being one-off investments). They might argue that it is commercially sensitive and have the scrutiny done behind closed doors but either way, the decision should be called in as part of the capital budget. I think most councils have a 2-week period of call-in.
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I have tried to stick to facts rather than opinions on the above. For clarity, I work with politicians from every political party. I am sure there are more local nuances that I am not up to date with.