I think that being In a different 'mode' on the mat helps me to treat my training as something special. Bowing to get on the mat, and then bowing to start practice, helps me to leave the trials and traviails of work/life/whatever behind, and concentrate on training. Bowing to the shomen feels like showing respect for the whole dojo, and there's not a handshake equivalent for that.
During flu season, I wish that I could bow rather than shaking hands all the time.
I probably don't bow the same way, and with the same mentality, as someone born and raised in Jajpan does, but it works for me anyway.
...is that different "mode" specifically tied to the act of bowing?
All cultures have customs that separate out various activities from each other.
The question is not whether or not some action of Japanese etiquette has a benefit or not, the question is whether that specific activity is necessary to the act of training in Aikido.
Let me say here that I have nothing against bowing, wearing funny clothes, or whatever - I do all of that on a regular basis (although there are plenty of times when I don't).
What I am
saying is that there is no intrinsic requirement
to perform those activities in order to be doing Aikido.