Toxic masculinity does play a shitty role in how men are expected to act and behave in society especially in conjunction with how patriarchal our society is.

Would I say it's oppression? Not really, it does push men to be emotionless slabs unable to express themselves properly.

But in my experience cishet people saying that cishet people are oppressed don't mean the toxic pressures of our current society and societal expectations.

Without knowing examples your acquaintance is citing it's basically impossible to have the conversation.

Does our society chew on cishet men? Yes.

Does our society chew on queer folk? Very yes, more so than cishet men as the chewing that it does on cishet men is usually also stacked on queer folk as well.