That might have owed something to the fact that Scotland's performance had left plenty of scope for improvement, because the new coach, like Jim Telfer before him, is one of those fellows who is never satisfied unless he has a lengthy to-do list on his desk. But with all allowances made, this was still an impressively solid and efficient display by the Scots, and laid a strong foundation for the rest of their summer tour.
An already daunting schedule - they will play in Canada, Argentina and South Africa over the next few weeks - would have looked far more harrowing had they launched the venture with the sort of drab defeat that has opened too many tours in the past. From an opening quarter in which the Scots punctured the United States defence and pulverised their scrum, that outcome was pretty much off the agenda.
The Scots did go through a listless spell in the middle period of the match, when the stifling conditions of a sultry Houston evening seemed to rob them of energy and will, but those qualities were restored when Stuart Hogg delivered a characteristically clever try in the 66th minute, a score that extinguished the defiance of their opponents in one fell swoop.
It was a moment of redemption for the full-back, who has lived under the shadow of his sending-off against Wales in March and in a swirl of rumours about his future. Hogg had made a bad error under a high ball earlier in the game, so the confidence with which he plucked a crossfield kick from the air and raced 65 metres to score was even more uplifting.
"He is a class player," purred Cotter. "He has probably had challenges in his life like great young sportsmen do and he is dealing with it. He did well for us and I am happy for him."
Hogg was far from the only source of comfort for the new coach. Finn Russell, the debutant fly-half, produced a game of all-round assurance and maturity, while Gordon Reid and Blair Cowan, the other newcomers in the starting XV, fully justified their promotions to Test level. Overall, there was a sense of purpose and direction about Scotland, in stark contrast to the shambles they had been in their final outing of the Six Nations.
Granted, they will seldom have such an easy time in the scrum as they did for the first 50 minutes of this game. The USA fielded some strapping lads - their energy in broken play caused the Scots problems at times - but the governor of Texas would have been entitled to declare their set-piece an official disaster area. It crumpled, it collapsed and it caved in, and it was a wonder that Scotland only took one try from that area of the game.
It is usually wrong to single out individuals in that area, but the quiet fortitude of Geoff Cross, the tighthead prop, deserves acknowledgement. Cross's father had passed away earlier in the week - the Scottish players wore black armbands as a mark of respect - and Cross's decision to play was courageous and entirely his own.
"He wanted to stay," said Cotter, who was clearly moved by the situation. "It is up to Geoff. He decides. There is no pressure on him to stay. You only get one mum and dad."
The US scrum was shored up when Eric Fry, Cross's opposite number - he plays for London Scottish - was removed at half-time, but the Scots were still dominant in that area. They were also far sharper out wide than the Eagles, with the back three of Hogg, Tim Visser and Sean Maitand offering reminders of their class.
Visser was sharp in support of Greig Laidlaw's break when he collected Scotland's opening try in the 15th minute. That brought his strike rate up to seven touchdowns in 13 Test outings, a return that few Scottish wingers of recent vintage have ever come close to matching, but it could have been more spectacular still had he converted even one of the three other chances which came his way in the opening 30 minutes.
Instead, it fell to the Scottish pack to extend the lead created by Visser's try and a conversion and two penalties by Laidlaw. As the half-hour mark loomed, Scotland set a series of scrums deep in the Eagles' 22 and went about their brutal business. One scrum led to a yellow card for USA loosehead Olive Kilifi, and the next brought a penalty try. At 17-3, the only question left concerned the scale of Scotland's victory.
Or so it seemed. Strangely, the 14-man Eagles looked a heck of a lot more impressive than the fully staffed version and they enjoyed their best period just before half-time. Captain Todd Clever even got the ball over the line, but the video official ruled that he should have released the thing in the tackle a moment beforehand.
The Eagles were also the better side throughout a third quarter in which the Scots became sloppy and appeared to be suffering in the heat and humidity. Hogg's try, though, extinguished home hopes at a stroke.
So a lot of boxes ticked at the dawn of the Cotter era. Scotland headed for the cooler climes of Canada last night, but they went with a warm Texas glow inside.