Totally agree with Globally... administrative duties while taxying out should be kept to a minimum.
But I also blame airports such as KORD (Chicago O'Hare) for contributing to an unnecessarily high workload because they don't issue your departure runway until you are with "metering" control, engines running and ready to taxi. The ATIS offers numerous departure runways, and if you guess wrong, now take-off data has all got to be re-calculated. If you don't start taxying immediately, expect to get flack from ATC.
Frequently during the 737 Classic days, I've seen the FO heads down with runway analysis manuals out while I've been steering the old beast around a taxi route I wasn't expecting. Sometimes I wouldn't start taxying until we got things somewhat organized. It's no better with the Airbus technology ... departure runway to change, new performance data required, briefings to give, etc.
ATC must have an idea what runway we are to expect well before we start to taxi ... why not let us in on the secret?