In "set of reasons that" what does *that* modify?

There is nothing grammatically wrong with the sentence. Let's parse it logically:

There is a set.

The set contains reasons.

The reasons are compelling.

The grammar tracks the logic:

There is [singular copular verb agreeing with singular predicate nominative set] a common set [singular predicate nominative] of reasons [prepositional phrase modifying set] that [beginning of a restrictive clause modifying reasons] are [plural copular verb agreeing with the actual subject, reasons] compelling.

Simply put, the set isn't compelling. The reasons are.

TITLE: Should a verb agree in number with a collective noun or its subparts?