I actually quite like Thomas Kochs. My fiancé worked in the management team with him at Cafe Royal, so I can assure you that you should not look too much into his short stay there as a blemish on his CV, as that property was and likely still is a sinking ship. The only mistake was thinking he had a big enough plug.
I think it greatly depends on the GM as to the impact they have. Some are great at marketing, so leave the running to the Hotel Manager; some are better with logistics, others better at improving the bottom line, others better at creating a guest experience. I've met the existing GM of The Connaught less than 5 times in 3 years, and I used to stay 50 nights a year. He joined at the same time as the new owners took over, which makes it hard to tell who had the biggest impact, but there has definitely been an overall cost cutting, reduction in amenities, staff retention and service. A different owner with a different vision brings in a GM to execute it. Who is to blame?
I absolutely hated my experience with the GM at Amangiri, but it didn't stop me going to Amanpulo when I discovered he had moved there. Unless he's a tyrant to staff, it hopefully will not seep into them dropping standards. Most of the time we don't really know whether it's the owners message, GM or some saintly staff who are responsible for the service standards, so I tend not to focus too much on it. There are some exceptional GMs that I would follow to another property, whilst the majority are neither exceptional or dross and I pay no attention to. Similar to almost every industry, including sports.
I feel as a guest the biggest impact we feel from the GM on a short stay is their actual physical presence around the hotel. It's always a pleasure to have the GM come greet you, chat to you and check in with you, yet in a 200+ room hotel you're unlikely to ever see him anyway.