Agreement problem in expressions like 'not only A but B'

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They say there's a regulation that requires an agreement of 'number' and 'grammatical person' according to the latter pronoun (B) of the expressions, such as, 'both A and B,' 'not only A but B,' 'either A or B,' 'neither A not B.'

But in case of 'B as well as A' because it is another alternative of 'not only A but B,' we have to make it agree with B in terms of 'number' and 'grammatical person.'

And they also list a number of sentences to show that rule.

* Not only the children but also the mother is sick.

= The mother as well as the children is sick

* Either your brother or I am mistaken.

* Neither your brother nor you were to blame.

I somehow have been thinking that the above rule is somehow bullshit.

Because I cann't find out any good English grammar book - such as 'Practical English Grammar by Michael Swan' and some other prestigious books - talking about the rule and some proper example sentences.

Am I right?

If I got it wrong, can you show me the rule - if you give me the reference book title it will be very much helpful - and some example sentences?