What are the advantages of squawking 7700?
If you make a radio call, unless you are on 121.5 (or 243 military), then only the station you are talking to will initially know about the emergency. Initial calls should always be with the unit you are working with unless you are VFR.

If you squawk 7700, then all stations in transponder range, including possible airborne stations such as AWACS and SAR will know right away that you are in trouble, with position, altitude and possibly aircraft type and flight number.

If VFR, you might not be talking to anyone and may not have the frequencies of nearby units easily to hand.

Squawking 7700 is definitely a good thing.

You mentioned being busy.  The mantra drilled into everyone is "aviate, navigate, communicate".  In most circumstances, fly the plane, fly it where you want it to go, then tell someone about it.  Squawking 7700 will be one of the last actions carried out.