And just to make it clear: there are a lot of ethical issues around the idea of lobby groups of any kind, in terms of a democratic state, but not really within the scope of this thread.
But relating to some of your questions above, there are two kinds of groups that are relevent.
Some are cause specific, like Greenpeace. They are supported by people who agree with what Greenpeace believes, and they lobby to achieve their aims. In fact, it is the members who tend to set policy direction, and people who disagree will tend to leave the group. They are arguing that what they believe is best, but they are only representing their own membership.
With identarian groups, like Stonewall, they are typically lobbying on behalf of an identarian group. Maybe gay and lesbian people, maybe women, maybe immigrants, or Muslims.
The problem is that in most cases, not all these people are actually members, and may or may not support what the group is advocating. So the group is in effect claiming that they should be accorded a status and importance reflective of the whole population they say they represent. and they also directly or by implication say that other viewpoints about said group are not valid.
This is why Stonewall is so upset about the LGB Alliance. What is reveals is that SW, while it may represent the views of certain people who are gay or lesbian, it does not in fact represent the views of that group as a whole - it is very much a partisan organization. If people realize that, it loses a significant element of it's political power.