I'm not a fan of a lot of the examples here for feedback up or down.
> “We may need to give even more guidance to new hires.”
Up or down, "we may" and "even more" are weasel words that weaken what you're saying. It's trivial for someone to interpret that as "they think maybe we could do more, but they seem to also see that we do a good amount already."
> “I used to struggle with this, and when I tried X, it really helped.”
This one I think is good for managing down; bad for managing up. It's less weasel-y but it risks coming off very aggressive - "I already figured this out, what is wrong with you?"
> “The team made amazing progress when we all focused on the website update last month. It might help to have one or two clear priorities for the team this month that everyone can rally behind.”
You have a great example, "it might help" is again weakening your POV compared to something more like a direct "what are the top priorities this month that we can all rally around?"
> “When we were able to dedicate that first week to training Steve, he got up to speed pretty quickly. The bit of upfront time seemed to have paid off, and taking a similar training approach for our next hire could help them ramp up just as fast. What do you think?”
Here we've taken a lot of words to state the obvious, which IMO both runs the risk of losing the urgency in the verbosity and coming off as pandering and over-explaining the obvious.
I think "what are your thoughts on" and "one approach might be" are better in both up-and-down directions as long as associated with a clear specific "here is something I noticed that I think is sub-optimal" situation.