Homosexual acts are legal - and have been for nearly 30 years. The Church of England, along with most of the other mainstream denominations, has concluded that homosexuality is an identity not a deed, that gay is something that you are, rather than something that you do - and a state of being cannot of its nature (and God's nature) be "a sin". (That some gay people would dispute this definition is irrelevant in relation to whether or not the Church of England should allow the meeting: its own position and definition are what count here.)
Despite these facts, rampant prejudice - in the forms of increased danger of violence, employment discrimination, housing restrictions, permitted public behaviour, access to children, to name but a few - continues. Daily, through public vilification, negative representation, and complex disingenuous lies, non-gay people have their prejudices fed and their desire to hate encouraged by our media, our heterosexist culture, and our Government's policies.
This causes real mental distress (the vulnerability to depression, self- harm and suicide of young gay and lesbian humans who are very often wantonly deprived of access to those who could help them is well documented). Homosexuals - quite separate from their own actual sexual practice - are treated as and feel themselves to be marginalised. This makes lesbians and gay men, as individuals and as a community, a group who have a specific, gospel-based, Jesus-led, call on the churches' charity and support.
The whole theory of sexuality is currently an issue of interest to the wider Christian community: all Christians have an obligation to discuss, debate and consider it. A group of Christians who are personally concerned with this pressing question wish to come together and do so. LGCM is a Christian organisation. Its members therefore are a part of the people of God, the body of Christ, the Church. It is our cathedral as much as anyone's.
Simply, the burden of proof lies not with LGCM but with those who object to the LGCM gathering, because the only "reason" I can think of is that there are people who cannot tolerate anyone else having prayer and fun from which they have let their fears exclude them.