Do you think mental health is just as important as physical health, or even more important?

Of course. There is a relationship between physical health and mental health. It is documented that each can affect the other. They cannot be separated. Both are most important.
It's all important. Mental & physical health interact with each other, for better and worse. I'll say something else that others may consider outrageous: spiritual health & balance underlie it all. When a person becomes "ill, " multiple levels of intervention (physical, mental, emotional, energy, etc) may be needed. All are important. It's also possible to heal, without one's condition being "cured" or eradicated.
Yes. Why do you ask? One of the difficulties is that there are even more charlatans in "mental health" than in other health promotion. Really promoting mental health involves helping people out of impossible situations, medicating for brain disease, and helping folks who've survived under horrible conditions relearn behaviors and attitudes. It's important work and good outcomes aren't always possible.