[Their insistence that the meetings should be held at lunch-time] angered [the staff]. (correct)
[Their insistence] was [that the meetings should be held at lunch time]. (incorrect)
[His fear that he might lose his job] was [increasing]. (correct)
[That he might lose his job] was [increasing]. (incorrect)
[His fear] is [that he might lose his job]. (correct)
From Collins Dictionary, insistence is
- the quality of being insistent
- the act or an instance of insisting
Some dictionaries list demand as a synonym for insistence, but it is easy to confuse "the act of demanding" sense of demand, which is similar to insistence, with "the thing demanded" sense of demand which is not synonymous.
In #1 "insistence" angers the staff and the subordinate clause describes the kind of insistence. It doesn't equate the insistence with the policy of having meetings at lunch-time. This is a correct construction.
In #2 insistence is the subject of the clause, so the sentence says "insistence" is "meetings should be held at lunch time". That isn't valid because the "act of insisting" is not the same thing as "a policy". You could say something like:
[Their insistence that was a source of anger for the staff] was [unreasonable].
This is correct because "the act of insisting" can be "a source of anger". Insistence is the subject of the clause "that was a source of anger".
In #3 "fear" is increasing, and the subject of the clause describing the type of fear he has. That is correct.
In #4 the sentence doesn't have a subject that the clause "that he might lose his job" describes, so the sentence is a fragment and not correct. @F.E. mentions in the comments, a way that the sentence could be constructed to be correct is
[That he might lose his job] was [his biggest concern]
In #5 "fear" equals "that he might lose his job" and that's OK because "a fear" can be "losing your job". Fear is the subject of the clause.