Thanks for posting that link - I hadn't read it before and found it really interesting. I've never identified as a liberal feminist (my early exposure to feminism was before that really got going in a mainstream way, I think); but I think that's also because the version of feminism I absorbed was also very influenced by Marxism. Radical feminism takes a great deal from the insights of Marxism (not that Marxusm has always been hospitable to feminism, but that's another story).
But of course liberalism per se is quite hostile to Marxism, because it doesn't fit with the pre-eminent emphasis on the agency of the individual as a primary force. So it would make sense that liberal feminism is an individualist feminism, essentially of the status quo. It doesn't really recognise the analysis of class that is the focus of Marxism - and maybe that is the central problem. One of the key tenets of Marxism is that ideological discourses work to persuade and cement people within their class position by making them complicit with their own oppression. In feminist terms, this means that women as a class often subscribe to the things that harm them. Women's choices are not always feminist ones, because they take place in a structure of oppression which works to ensure many women make choices that are actively harmful to the interests of women as a class (though those choices may be variously good or not for those individual women).
If anything, intersectionality ought to teach us exactly this: that I as an individual might make a choice that works for me or even advantages me, but that this choice might well also be thoroughly within and contribute to a structure that harms women overall. But in liberal feminism liberalism can't recognise the second part of that, because the key thing about liberalism is that the individual is never blinded in the choices he or she makes. So you get intersectionality being instead a kind of empty gesture, a round of privilege checking that doesn't really go anywhere.
We are really resistant at the moment to recognising that classes of people can be complicit in their own oppression; NOT because they actively choose to be so, but because they are ideologically blinded to the operations of privilege that work upon them. Marxism aimed to give people the tools to see where the economic interests of capital were acceded to by workers because they were ideologically naturalised as right and proper parts of life. Radical feminism aims to do that too for women - to be able to say that some choices made by women are not actually freely chosen, and even if they are are not free of harm or the structure of oppression. (I might not be racist myself, but I might make choices that are fully in keeping with and even help shore up a racist social structure, for example.) Liberalism in all forms is very resistant to this being pointed out.
Just musing on these things...sorry for the long rant!