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The potential customer searches for specifics so they should be able to see a page suited to their search. You only have one chance to keep many customers.
Analyse the performance of their homepage currently. Do most visitors leave? If so then there is your justification. If a good proportion buy then a PPC campaign could be the answer.
Sorry this is short
(The client's attitude is the sort of controlling behavior that really will turn many customers off.)
You should have some of your interior pages performing as indepentent marketing agents in the SERPs bringing in highly targeted visitors...typically better conversions off interior pages...
Obviously, your client's perception is about their brand and what "they think" the home page offers the targeted visitors...
They may have several product/service offerings running concurrently (most likely) and they want all traffic to see this diverse range...(not good conversion...but good confusion)