Pretty much what Katsunami said, with two additions:

It is possible to make a PDF that will be readable on 6" readers, but chances are that you're going to have to reformat the entire document (e.g. to use larger fonts) so you may as well go to ePub. Another drawback for PDFs formatted for a 6" reader: it won't look very good on larger ereaders and tablets, tiny screens for phones, and huge screens on computers.

If it is a text heavy book, expect customers to turn down PDF formatted books (or to be angry if they accidentally buy a PDF formatted book).

Technically your client is correct: most, if not all, ereaders can handle PDF files. They do handle them to varying degrees though (Kobo, not so good; Sony, fairly well). Yet that doesn't mean that it is a good format to publish the book in unless it already uses a complex layout.