Practically though all the roles available only take the secular into account. In other words unless there are positions based on spirituality as there are say for LGBTQ issues in local government only atheist, materialistic interests will be served.
But that is a different question - whether a priest might find the role (for example local councillor) attractive and important, rather than whether they should be permitted to hold that public office.
That said, I'm struggling to see why serving the public in the role of a councillor wouldn't be appropriate or relevant to a priest - indeed many of the skills and roles necessary are rather similar to that of a local priest - interactions with and supporting a community, which in the case of a councillor would just be somewhat wider than in the case of the role of priest.
And I can't see how the primary functions of the council - e.g. housing, social care, planning, service delivery wouldn't be seen as important and relevant to a priest.
And by the way I doubt you will find a portfolio holder function for LGBT issues most local councils - there certainly isn't on mine -
http://www.stalbans.gov.uk/Images/FINAL%20AGREED%20Portfolio%20Grid%20July%20onwards%202017-18_tcm15-54142.pdfIt is true that the council needs to take account of its legal obligations on equality and diversity, and has a section on its website for this:
http://www.stalbans.gov.uk/council-and-democracy/equalities-diversity/In which all protected characteristics are covered, with none are given any more prominence than another:
'The Council is committed to promoting equality and diversity and meeting its duties as a community leader, service provider and local employer.
Equality Act 2010
The Equality Act 2010 replaced a large number of previous equality laws with a single Act. This has helped to simplify the law, remove inconsistencies and make it a lot easier for organisations and individuals to understand. It strengthens the law tacking discrimination and inequality across all of the ‘protected characteristics’ set out in the Act.
The Council must have due regard to the need to:
Eliminate unlawful discrimination, harassment, victimisation and any other conduct prohibited by the Act.
Improve equality of opportunity between people who share a protected characteristic and people who do not share it.
Foster good relations between people who share a protected characteristic and people who do not share it.
Protected Characteristics
The ‘protected characteristics’ set out in the Act are:
age
disability
gender reassignment
pregnancy and maternity
race – this includes ethnic or national origins, colour or nationality
religion or belief – this includes lack of belief
sex (men, women, boys, girls and transgender)
sexual orientation
The duty applies to marriage and civil partnership too, but only in relation to the first aim (to eliminate unlawful discrimination).'
So religion is just as important and prominent in the equality and diversity agenda as sexuality.