In 2016, the House of Commons' Women and Equalities Committee issued a report recommending that the Gender Recognition Act 2004 be updated "in line with the principles of gender self-declaration".[133] Later in 2016, in England and Wales, a proposal was developed under Theresa May's government to revise the Act to introduce self-identification, but it was dropped in 2020 after opposition. Instead, Boris Johnson's government reduced the cost of application to £5, and the procedure to apply was moved online.[4][6]
The crucial bit is in bold.
The Conservatives (or at least some in the party) are embarrassed by their previous stance on gay rights in past decades. Public opinion on homosexuality has shifted significantly and same-sex marriage was eventually successful. As a result, many politicians across all parties think that 'trans' and 'gender identity' issues are exactly the same as the gay rights movement. They are scared of being called homophobic and gender identity activists love to exploit this fear.
The advice in the Denton's report tells gender identity activists to massage the egos of all politicians - tell them they will be on the right side of history by supporting self-id. Lots of lobbying in the background away from public scrutiny and transparency - attaching the controversial trans stuff to issues that have much wider public acceptance like gay marriage.
Most politicians love the thought of 'making a name for themselves' and self-id initially looks like an easy win. Like other parties, the Conservatives jumped on the trans bandwagon because they thought it wouldn't take much effort to implement self-id and it would be an easy win to make them look progressive since the gender identity brigade keeps telling everyone it's no big deal.
The important bit is that even though the Conservatives thought self-id was an easy win, they responded to opposition. Some have clearly grasped the harms to women and children emanating from this ideology. And overall the party is SMART enough to realise that self-id isn't a vote winner. Staying in power is important to the Conservative party and I can't fault them for that. It's what I'd expect from any political party of substance. Once they realised self-id was going to be a political shitshow, the leadership was clever enough to change course. Reducing the fee to a fiver and switching to online applications was a stroke of genius. How are the gender zealots going to screech about the 'trauma' and 'degradation' of such a process now? They will have to find another hyperbolic angle.
Yes there are still politicians within the Conservative party that want to push for self-id, but opposition to them is getting vocal.
I have no doubt that if Labour were in power, self-id would have been steamrolled through. Leadership appears to be consistently spineless, unable to push back against zealots even though it is clear that public support is not there for these wacky ideas. If this is what the party is like without power, it doesn't inspire confidence in people that they can handle the pressures of being in power.
Labour doesn't come across as a serious party. Much of UK voters are fairly centrist and socially conservative. The public perception is that the Labour party is focused on trendy activism that is popular with its niche, affluent supporters and prefers 'purity' over pragmatism.
Research repeatedly shows this is the public perception but the Labour party just cannot acknowledge this for some reason.
They are hamstrung by their over-reliance on divisive identity politics that spans much more than just the gender identity stuff, and the Conservatives are smart enough to exploit that for their own benefit.