The term of art for this practice is “skip-level 1:1s”. It’s not ubiquitous but many think it’s a good idea, particularly when stepping into a new role/org. Obviously it can end up being a lot of meetings so it’s typically on a quarterly or less cadence.

Because of the branching factor it’s not feasible to do this for every bit of comms.

Yes, this glue is important. E.g., the VP dropping by the EM's staff meeting to give a tailored version of the message and do some Q&A, and the VP doing skip-level 1:1 meetings to get the perspective as seen by the EMs. If you build kind of a mesh of redundant communications, you can better course-correct (which also means correcting the original message after observing its actual effect on the team).

An additional benefit of 2-level comms is that you get send some slightly-irrelevant information, but it's clear you aren't expected to read it. This gives you a passive awareness of some of the other stuff that's going on in the organisation, and who you can ask about it.