I share your feeling. Here are some answers I've come up with for myself.

If you want engagement... Your manager got you to share it by directly asking you to. You will have better luck receiving feedback if you follow that lead. You asked them to look at it, and got discouraged by the lack of any reciprocity. Ask for engagement, if that's what you feel like you want. One effective way for me to do this is to share my assumptions. :)

If you don't want to feel like you are playing that "importance" game... Play it on behalf of your team, instead of you. Give kudos to everyone around you. And be genuine about it.