Every time someone says we should centre women in our thinking and other groups can take care of themselves they are making the argument for other groups to take the same approach.
Yes, and why is that wrong? Besides, that's not quite the argument we are making. We are saying that we should centre women in the women's rights movement in response to the ludicrous claims that if our feminism doesn't include fighting for the rights of males who identify as trans it's not feminism. Our argument is that feminism should centre women and males who identify as trans should have their own rights movement.
But if they are going to lobby for a particular right for men, because having thought about it they think it would be good for them or solve some problem men have, they need to be considering what the wider effects of that policy would be. If it would mean more children in poverty for example, or less recourse for domestic abuse, or create bias in career hiring, that's a problem and they maybe should consider that their idea is unfair, however good it is for men as a group.
Rights movements, whoever they centre, have only one function: to advocate for the needs and rights of the particular group of people they focus on. The Royal Association for Deaf people has no obligation whatsoever to consider what impact something they are calling for has on others, not even other protected groups. They can legitimately draw up policies they'd like to be implemented and focus entirely on deaf people.
I think your example is probably not working too well, because men as a class are the default that underlies policymaking. In our patriarchal societies they are an oppressor class. Those campaigning for the rights of the oppressed have to fight to be considered at all. Any additional rights or privileges implemented for the benefit of men would in all likelihood negatively impact on others, which is why we would expect them to consider the impact on disadvantaged groups. But they wouldn't. No oppressor class does.
However it does not follow that campaigners for the oppressed or disadvantaged can only legitimately advocate for their needs after considering the impact on all others, which would then obviously also include the oppressor class.
For starters, it would make effective advocacy all but impossible - most of these groups are run by volunteers with no funding. They also often understand only one issue deep down to the minutest detail as well as the latest developments - their own. Often they genuinely cannot even know what the impact on other groups would be.
That is why it is vitally important that all organisations and groups who advocate for protected groups are allowed to publicly make their case and to take part in any debate that arises where rights may come into conflict. Because then one campaign group's proposal would be reviewed by other groups who might respond by highlighting any negative impact on them and maybe making alternative proposals.
And that is why it is vitally important that we have a legal framework - the Equality Act 2010 - which helps policymakers and legislators to consider the merits and drawbacks of any policy proposals or legal reform proposals put before them by campaigners.
Of course, sometimes the system fails us, but that's not because campaign groups focus on their own campaign. That's what happens when only one side is allowed to campaign freely and given access to policymakers and legislators. As happened here.
We wouldn't be any more successful, only less, if we couldn't simply point to the negative impact on us of their proposals, if we couldn't simply advocate for what we need instead of having to do the heavy lifting for the other side as well. It's not our job to find solutions to the problems males who identify as trans have because of how other males treat them. It's theirs.
Our job is to stand up for women.