I am not a teacher.
The rule is simply to look at what the subject of the verb is.
"Our customers and their success are top priority" is correct, and you were right that joining nouns with "and" gives a plural subject.
With, "Our top priority is our customers and their success" the subject is singular, "priority".
A common exception to this rule is that of mass nouns such as team/group etc. which can be regarded as plural even though they are grammatically singular.
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