My first contact with a GM is often after the stay. If I have an outstanding stay, I do my best to get the email address for the GM. I write a thank you note, calling out specifics of whom and what made the stay excellent. I think they hear complaints so often it is refreshing to hear from a happy guest. The GM always responds. This then sets me up should I want to let the GM know that I'll be inhouse again. Here's an example. I went to the Gritti Palace, Venice, for a milestone birthday and had a wonderful stay. I was solo and was treated beautifully by every hotel employee. I wrote to the GM to say thank you. He asked me to let him know when we were in house again. A couple years on I brought my husband for our wedding anniversary. The GM upgraded us to one of their named suites, an act of great generosity and well beyond anything I expected.
We met the GM at Caresse in May. It was very quiet at the hotel and he sought us out to ask about our stay. We had finished breakfast and asked him to sit with us at our table. We have a lovely conversation. We mentioned many things that worked well and a couple that didn't. An example of the latter was the lack of Turkish items on the menu. He had the chef set up a special four course meal for us and gifted the first and last course to us. He then chatted with us on departure. I think the conversations were as enjoyable to him as to us as we had a similar mindset about what a luxury hotel experience should be. A few weeks back when there was an earthquake in Bodrum, I dashed off a one line note of concern for he and his staff. He responded quickly, letting me know that everyone and the property were well.
I can remember two other specific instances where thru serendipity the hotel GM fell into conversation with us and we ended up having satisfying conversations about the hotel and their experiences. It's not something that can be forced but when a skilled GM engages, it creates a genuinely satisfying guest experience.
I recognize this is not to your point, KatW, but I hope it's of interest. I enjoy the questions that you pose.