Certainly, socialism hasn't significantly outpaced capitalism. You can point to pockets like east Germany, but overall the socialist block hasn't been great on the issue. For the examples that look good, (Cuba, and China's improving) I think the factor has to be time rather than socialism.

I have to disagree with the framing of this.

While it's much easier to measure economic gains made by socialist countries, it's far harder to quantify progress on queer liberation. There's the obvious elements like policy but policy is usually representative of an undercurrent of cultural and governmental attitudes towards queer people, but to quantify the progress made on societal values, acceptance, and inclusion as well as healthcare outcomes, crime statistics, rates of unemployment and homelessness and addiction etc. is very difficult as gender studies is still a frontier of social research even in the west today let alone the Eastern bloc countries or East Asia.

It's also important to keep in mind that a socialist government inherits the cultural and societal values of its predecessor state; these attitudes don't start from a default position or a blank slate. With that said, it's also worth keeping in mind that to argue that socialist countries haven't outpaced capitalist countries is to give most capitalist countries a headstart of at least a century but often far more than that.

We can look to countries like Poland, Hungary, and even the United States to see that queer liberation is also liable to backsliding so I also think it's a bit unfair in looking at "the west" because it's very easy to cherry-pick the most advanced countries while ignoring other countries that are backwards or even retrogressing.

Last thought - this is also complicated by factors like advances in gender care in countries like Czechoslovakia; while it's undeniable that Czechoslovakia was extremely advanced in matters of gender care it came coupled with inherent heterosexism which is easy enough to argue as being a backwards step but I think it's better understood as representative of an uneven development towards queer liberation. (See the work of Appeltova for more on this.)