“Don’t be needed” isn’t “don’t be valuable.” The EM should not be a bottleneck. The EM should be able to take a vacation without being paged. (So should anybody on the team!)
My teams would slow down without me because I can due process tasks more efficiently, but nothing demands me to be in the loop.
Someone I know used to be pretty senior at a major SV company. Over dinner one night, he told me that the CEO would take vacations with instructions to the effect of "handle it" if something comes up. (Assume it wasn't absolute but that was the basic gist.) Apparently, a new PR head came in and was like "I can't work under that condition" and quickly left.