FYI Illiberal values aka non-liberal values (clarifying because the I is hard to read) use the word liberal in the traditional sense.
Liberal values are liberty/freedom, consent of the governed, and equality before the law. All other liberal values build off of these three as a base. This implies that Non-liberal (or illiberal) values are the opposition of liberal values through censorship, gun control, etc like you mentioned.
Liberals in the modern US political sense refers to Neo-liberals. Neo-liberal and liberal are two very different things which is why the term liberal value doesn't necessarily correspond to neo-liberal beliefs.
Additionally, "the left" by and large does not support neo-liberalism. "The left" is violently against the aforementioned censorship, gun control, etc. Reading any socialist or communist literature will make this abundantly clear.
Examples:
- George Orwell on the Right to bear Arms: "The totalitarian states can do great things, but there is one thing they cannot do, they cannot give the factory worker a rifle and tell him to take it home and keep it in his bedroom. That rifle hanging on the wall of the working-class flat or labourer's cottage is the symbol of democracy. It is our job to see it stays there."
- George Orwell on Freedom of Speech: "Threats to freedom of speech, writing and action, though often trivial in isolation, are cumulative in their effect and, unless checked, lead to a general disrespect for the rights of the citizen."
- Karl Marx on the Right to bear Arms: "Under no pretext should arms and ammunition be surrendered; any attempt to disarm the workers must be frustrated, by force if necessary"
- Karl Marx on Freedom of Speech: "The absence of freedom of the press makes all other freedoms illusory. One form of freedom governs another just as one limb of the body does another. Whenever a particular freedom is put in question, freedom in general is put in question"
- Karl Marx on Freedom of Speech: "Censorship has outlived its time; where it still exists, it will be regarded as a hateful constraint which prohibits what is openly said from being written"
- Karl Marx on Freedom of Speech: "You cannot enjoy the advantages of a free press without putting up with its inconveniences. You cannot pluck the rose without its thorns!"
If you want I can dig up more quotes but those are the ones that were easy to fetch and any more risks turning this into even more of a wall of text.
My point being, your issues with "the left" are misdirected and are better focused towards Neo-liberalism and/or Neo-conservatism. "The left" does and has always been one of the primary guardians of liberal ideology. Hell "the left" is where a significant portion of the liberal ideology that the United States is founded on originated from.