Let's leave out the fact that President Trump continually rejected "the herd" strategy (while using him in the picture). He did this multiple times in press conferences.
It's his administration official making the statements in multiple emails over many days to the groups responsible for taking actions. It's his picture because he's in charge of the administration. It's a fair thing.
It could be that this official was out of sync with the goals of the administration as a whole. In which case, it should have been caught and then aligned with the overall consistent goals. That it wasn't, or wasn't aligned fast enough is poor management.
It could be that the administration doesn't have a unified goal, but independent actors with conflicting messages. This is poor management too. This leads to lots of problems as people try to figure out who to listen too.
It could be that the administration has a habit of making one set of statements in public announcements and a different one in implementation communications. From the outside this feels like poor management, but it could be brilliant. If the goal it look like one thing but do something different, then this structure is completely appropriate. But, then it's also completely correct for reports to call them on the differences between public announcements and implementation messages.
In any case, he's the leader of the administration and it's fair to use his picture when describing administration actions. No matter what's said publicly, those are still the actions of the administration as a whole.
Also, is it a huge revelation that reducing restrictions would lead to more cases? That's part of why I argued against the restrictions in the first place back in March and April; because, as soon as you remove them, the spread increases. My point was that "control" requires restrictions in perpetuity until vaccine availability.
This is still wrong, and there's still more tools that can be used instead of only the hammer of restrictions. It's a choice to not use anything besides restrictions for control. Obviously, if we're going to "do nothing", then cases will go up. That's wasting implementing other actions when spread is at a low enough level for them to be effective.
Finally, what he said about herd immunity by letting the young and invulnerable get infected was not scientifically incorrect. It would have required some way to completely isolate the older and vulnerable while 200 million+ got infected, which would be extremely difficult, if not impractical.
It would have required millions of deaths. What he's saying is that he's fine with millions of dead people as the solution. If millions sounds like exaggeration, then he was fine with hundreds of thousands of deaths. It's not really any better.
Isolate the infectious has always been the correct answer. There's less of them than any other group. Bonus, the better job done isolating them, they'll go down in numbers. The problem has always been how to find them faster than they change. The general restrictions have always been the same broad based untargeted way to reduce all interactions by isolating everyone to catch the unknown infectious subgroup. It's a good plan when community spread is high, since there's a ton of unknown infectious people. It's a poor solution as the number is reduced.