I was on the board and Treasurer of a 144-unit condominium association in Florida. Technically, you need to look to the contract to see what the managment company's duties are, but also look at what the management company has historically done for the association as IME most management company contracts are rarely granular enough to answer such questions, at least ours was not.
Prior to joining the Board, I would have thought that the management company did most everything and the Board's role was more big picture and strategic, setting the parameters within which the management company operated. That was not the case. The Board did a lot of things that I would have thought the management company should have done. Part of it was that we had a long-time rep from the management company that was generally incompetent so over the years Board members had picked up his slack. Also, that manager was a bit of a neanderthral with respect to technology so any practice improvements for the association fell to the Board members since this guy was clueless... he couldn't even run a Zoom meeeting without screwing it up.
For insurance, we had a relationship with a local insurance broker who shopped available companies for coverage at the best premium. We also had an insurance appraisal of our buildings annually or bi-annually, I don't remember which but I think annually, that was used to set the coverage limits. While I would have thought that the insurance renewal would be handled by the management company and served to the Board on a silver platter, that was not the case... the Treasurer was the key contact with the insurance broker for renewals.
I reluctantly ran for reelection not because I wanted more but because the alternatives were worse. When I narrowly lost I was overjoyed... I've never been so happy to lose something in my life.