Many researchers don’t want to do public engagement. Few consider it part of their core mission, many consider it a waste of their time. As for higher education institutions, they send out a mixed message: yes, please do it, but do it in your own time.
The problem is that public engagement is perceived as taking valuable time away from research, which is already compressed by teaching and administration. It also requires a considerable amount of preparation and even training to deliver effectively. So even when engaging with the public is unavoidable, because it has been worked into the dreaded “impact” section of a grant application, it is either done begrudgingly or palmed off to junior colleagues on the basis that it is all good experience for them.