I think you're close here, but I don't think that's the root of it. That difference in position is important, but it comes from even deeper philosophical differences. I think this boils down to the argument over positive vs negative rights (freedom to vs freedom from).
A worldview built on negative rights has to have a much smaller legitimate role for the government than one built on positive rights. In the former, the government can never make anything better, only potentially rectifying wrongs (usually via punishing the offender). Within that framework, the goal of the government becomes to rectify the worst of the wrongs while disturbing things as little as possible. A worldview built on positive rights inherently leaves a bigger space for the government to behave legitimately, because this worldview sees the government as having a role in making things better.
Of course, there are very few people who adhere 100% to either of these, so reality is a bit more complicated.
There is a section of the left-right spectrum that believes in equality of opportunity, but that it’s certainly not the leftmost people in that spectrum. The left pole is not “equality of opportunity for all” and how far left you are is defined by how strongly you believe in that.
I think it’s more the middle of the spectrum who hold the view and the left extreme is much closer to equality of outcome.
People who support police when police breaks laws are not pro-enforcing laws. People who make it increasingly hard to prosecute and investigate cop when those are suspect of breaking law are not pro-enforcing laws. People who make it legal for police to use escalating aggression are not pro law enforcement either.
These people are pro state violence people. Simple as that. Whether it is happening in Russia, America or elsewhere.
I think it's more that the left believes government has a responsibility to guarantee equality of opportunity, not equality of outcome in what is assumed to be a fundamentally inequitable system. The big philosophical difference to me is in what each side believes the proper role of government should be. The right (tending to be the side that supports the military, police, harsh criminal penalties, etc.) believes government's only legitimate purpose is enforcing the laws and defending the borders, in other words the exercise of violence, whereas the left also believes the government has a duty to provide for the social welfare of its citizens in exchange for the power granted to it by social contract.
And when the right says it wants liberty, what it tends to mean is that it wants liberty from the social obligations that the left tend to want government to support.