They could have avoided making any direct comment on effectiveness and simply stated that access by medical personnel needs to be prioritized since they're the most exposed group.

Even if we grant the premise, this is just such a dumb lie. It's universally obvious that masks work; you can look at pictures of everyone in other countries wearing masks, and the 1 second thought of "the virus has a hard time going through the mask" is substantially correct. How's anyone who realized that they lied about masks supposed to trust their other public health recommendations?

Perhaps, but it also pours gasoline on the flame of skepticism, once the cat is out of the bag and there's proof that they have lied. I.e. those decisions don't exist in a vacuum, they require (and burn) trust built in the past and decrease trust in the future. At some point it ceases to work and only makes things worse.