Can you cast and hold Produce Flame and then wield a weapon and shield?

The text of the produce flame spell reads:

A flickering flame appears in your hand. The flame remains there for the duration and harms neither you nor your equipment. The flame sheds bright light in a 10-foot radius and dim light for an additional 10 feet. The spell ends if you dismiss it as an action or if you cast it again.

Emphasis mine. The druid in our party asked our DM if he is able to cast produce flame in a dark cave, then opt not to attack with it, and wield a shield and sword. The goal is to continue emitting light, since he doesn’t have the light spell. He claims that since the flame does not harm his equipment, he is able to hold a shield or sword with the fiery hand with no problem.

Is this okay to do RAW, or are we missing something?

This is specifically not asking about casting spells with a sword in hand. He wants to draw his weapon after casting and holding the flames.

Related, but not the same question: 1, 2, 3.