In other words, is it too costly for PB to be an EB
Answering your second question first, PB's are EB's; they just provide an entire suite of products and services beyond just trade execution that an EB would provide. If you have an account with a prime broker, they can undoubtedly execute trades, and you wouldn't necessarily need an additional EB.
Why would a customer go to an EB assuming that he already is availing
services of PB?
The answer here may be more obvious than you realize. No broker, whether they are prime or executing, has an exhaustive list of every single potential counterparty out there. The more brokers you have, the more access you may get to liquidity from different sources. This is especially useful cases where you have an EB that specializes in trading certain products or securities that other EB's or PB's do not. For example, I am looking for a large number of shares in a company whose stock isn't very liquid. Instead of trying to execute this myself with a generic algorithm (VWAP, TWAP, or something similar), I decided to call my brokers to see if they can source some large blocks for me. Traders at my PB and my EB, separately and simultaneously, are sent requests to look for blocks of shares on my behalf. There is no telling which trader, if either, will be able to source the shares for me but, I have a better chance with 2 traders on the hunt than 1.