Reasonable comments. But the problem was - as supported by the quotes in the OP - that many people have been taken an unreasoned approach to this issue. They have claimed it is obvious, duh, that maths is true, objective, independent of human minds.I tend to believe that any time we use language or communicate, there is bound to be subjectivity involved.
Anyhow, a more appropriate question might be "must axioms be objective". I don't think they are required to be.
We can whittle and pick away at the different perspectives of axioms and whether they are subjective or objective, but in the end, these are ideals, like a perfect conductor or a perfect insulator... they likely don't really exist.
Instead, things can only be compared: more subject or more objective. I think mathematics is one of the more objective things out there (not to be confused with perfectly objective).
So it is useful to get to the root of the issue.
A second more important reason is that once it is understood that axioms involve choices, then we can consider how those choices are typically made. What is the thought process by which good and useful axioms have been developed? This opens up a discussion of the theory of axiom construction. Whereas if you believe axioms are found rather than developed, then such a conversation, such an epistemological self-examination, seems pointless.