I think that there are a couple of major problems, that the left must eventually overcome. What are these issues?
Well, for starters the left tends to suffer from the microbe of economization, which it probably inherited from the capitalistic imaginary.
The second issue -which i also think is quite important- is that it really doesn't deal with anthropology that much. The right sees only individuals, the left expands the mode of analysis to systemic/ institutional levels, but in the end both of them refuse to access to a sufficient degree the anthropological dimension.
Lets start with economization. I think the major problem of society is mainly a political one and not an economical one. Of course the economy and politics are interconnected, and one could say, that of course it is also a political one, but from what i can see, the left gives too much emphasis on the economic/social/ productive sphere, while it tends to ostracize the political life.
For instance, Michael Albert, an anarchist, has written a book, about giving a description for a post capitalist society. Inside it, he gives an extensive and detailed description about how workers would organize their working environment and how they would interact with consumers to distribute goods according to their needs.
However, there is very little or no emphasis given at all about the pure political arena. How would society be organized on political grounds, outside the working environment? The left tends to sideline this sphere in favor of the productive one. As a result historical socialism reproduced a monarchic/oligarchic political typology, where a few ruled over society, mirroring in some ways the liberal societies on that department.
There are a few thinkers like Boockhin and Castoriadis, who escaped from the concept of economization,and who think that the problem is mainly political not economical.
Lets now move forward to the problem of anthropology. The left tends to think that humans are born relatively decent, and develop problematic tendencies because of the functions of opressing institutions. Of course, to some degree, this is a valid observation.
However, they really dont analyse that deeply the human mind, its character, its ethos etc. They also tend to be anti-tradition, even though there are "conservative" traditions that can cutlivate responsible people that would be able to manage their own affairs.
This is especially true for the anarchist wing of the left. It is even more problematic in their case. The reason that it is even more problematic in their case, when compared to a communist, is that anarchists tend to support a more immediate or pure form of collective self management.
If you don't have a proper anthropology, how do you create a human typology that is able to self govern democratically itself and its society collectively? The demos must be made by a body of citizens who can face hardships, who have a feeling of responsibility, who come towards through face to face relationships with people they dislike etc. in order to make decisions for the common good.
The left talks about values such as cooperarion and mutual aid, but it lacks a rich and solid tradition that can cutlivate this human typology, one that will be able to reproduce and preserve a proletariat democracy, or a collective self-governance.
In short, a mature and highly demanding political typology requires an analogous mature human typology to fuel itself. Capitalism requires an immature typology to preserve itself, because it treats people simply as consumers, tourists, airport travellers, cosmopolitans etc.
This must be reversed, we can't accept ourselves as we are.
These two problems are interconnected, meaning socialism -as is largely understood today- doesn't escape outside this consumeristic paradigm. It might aim to give people goods according to their humane needs instead of purchasing power, but consumers they remain nonetheless.
It doesn't aim to produce autonomous citizens that can self-govern their demos collectively.That need doesn't arise because the even more mature techno-productive base of future communism, will be able to take care of people needs well enough, to the point that face to face participation for the common good will not be a priority or even a necessity.
The logic is that if we develop a better machine, that can give us a richer materialistic life then we should do it. This however will keep producing human typologies similar to the western ones today. Therefore the anthropological problem is not solved and neither the political one.
You can have an advanced productive base, which serves the needs of humanity, and that system could still cultivate self centered, hobbyist human personalities that can't relate to one another, and it could still have an oligarchic character as a matter of design.
The problem of freedom and that of community and relation doesn't get solved.
The left must realise that the political typology is the most important problem, otherwise it will keep developing monarchies or oligarchies in its attempt to overthrow the oligarchy of capitalism.
And it needs to take more seriously the anthropological dimension. That is my take.