There are many small problems that per company and aircraft manufacturer directive in the flight manual/QRH that demand pilots land at the nearest suitable airport should they encounter that kind of EICAS message.

Declaring an emergency and bringing out crash fire rescue is often a precaution both for safety and liability. I think if lawsuits were less common, airlines would call out fire rescue less often and only for more serious issues.

I seriously doubt that the rudder was wobbly. That is an extraordinarily dangerous situation and there is no way a passenger would be able to tell that anyways. If the rudder is wobbly, It is highly doubtful you will be able to safely make a landing in any kind of crosswind and turning and controlling the plane would be a serious hazard or impossibility.

Emergencies are a busy time for pilots. They need to notify dispatch both in flight to get the release amended and also on the ground to de-brief them on the event. They need to contact maintenance control as well as local maintenance. On top of all this need to do their normal flying and ground operations procedures. Maintenance procedures involve a lot of trouble shooting. Maintenance will initially see what the crew can do to trouble shoot and see if the problem may go away once they re-boot everything. If this doesn't work, mechanics will come on board and the pilots will need to explain to them what is wrong and work with them. On top of all this is all the paperwork pilots needs to do both in writing up the log books and for the incident reports the FAA and the company need from them.

Pilots also are told to put maintenance issues in terms that people understand and to not scare them. A problem that may be routine to airline operations may be more traumatic sounding to the less experienced flyers. Less information can be better than more. Pilots can get themselves in trouble by saying too much to an audience that is lacking in aviation technical knowledge.