Whilst ‘technically’ correct to say the chair/Mayor is ‘just another Cllr’ that is not a very accurate representation of the reality for many town and parish councils. They are first among equals and that of itself bestows certain ‘influence.’
Then there is the chair’s allowance of course - unilateral authority to spend and not subject to further approval nor account in audit.
Actually, the chair / clerk combo exercise considerable (and certainly not always equitable) influence - and therefore authority - over the actual written agenda, the tone, tempo and focus of the meeting AND how the minutes represent the record of the meeting.
How many posts and comments do we see from Cllrs querying and questioning chair and clerk ‘ownership’ and influence over the agenda.
An effective chair has massive influence over a council.