Why? (Seriously - you seem to be assuming the answer here.)

Would the same statement be true for food and shelter (also necessary for life)? Should the government be the provider for all food and shelter in order to avoid additional steps?

Because a person is capable of determining good produce from bad produce and make informed decisions when purchasing groceries.

The same is not true for healthcare. Even when purchasing shelter, people are not sufficiently informed to be able to make a good decision, hence the existence of electrical, plumbing, and structural codes and inspections.

With healthcare, people are extremely uninformed. The costs are extremely high. The government doesn’t necessarily need to vet the doctors’ diagnoses and prescribed remedies, but they could. So could insurance companies, which is what currently happens in the US.

This is a bit chicken-and-egg. People are uninformed about cost and quality of healthcare because the third-party payer system obscures detail (often intentionally).

Spend some time uninsured and you become a very discerning consumer.

Further, for every service that requires healthcare advocacy (often provided by family members, btw, not professionals even in existing system) there are probably >five that are commodity services (labs, prescriptions, diagnosis, imaging, etc.)

Maybe. That would have economical benefits also by lowering crimes related with lack of food or shelter (therefore releasing money to spend in other things and contributing to social peace). The plan would deserve a closer look.

The difference between an apple being stolen by a homeless and an apple being given to a homeless, is that you don't need to pay for fixing the broken window.