It's definitely a big plus to me when the GM is engaged in FOH activity such as greeting guests. It's not the introduction or chitchat I care about, per se, though. It's the fact that the GM is directly observing service delivery and customer experiences. Generally speaking, such a GM is going to be more on top of customer satisfaction problems than one who hides in the back office and knows only what his/her department managers report. Thus I'd expect to have better experiences in general at a hotel with a visible GM.

OP: I'm curious if you have a particular reason for asking this question. Do you work at/with a hotel? Have you recently encountered a GM taking a visible presence at one of the hotels you've visited? How did/do you feel about it?