Joe Root spoke to the media at Lord's on Tuesday afternoon, having had a "strong" conversation with the England players earlier in the day. England captain Joe Root says he is "hacked off" and "disappointed" at the incident involving fast bowler Brydon Carse at a Derby nightclub.

Carse, 31, was dropped from the squad for the second Test starting against Pakistan this week. Images that appeared on social media on Sunday showed Carse, 31, in handcuffs outside a bar in Derby on Saturday night, and he was briefly arrested and subsequently de-arrested.

"If I'm being brutally honest, I'm really hacked off, I'm really disappointed," Root said in the opening to a news conference. "After last week's performance, we're sat here and we're talking about something that is not related to it at all."

The England and Wales Cricket Board (ECB) has referred the incident to the Cricket Regulator, and Carse was replaced in the squad by Sonny Baker, with the Test starting at Lord's on Thursday. Carse was not on England duty at the time of the incident as he was released to play for Durham, but he is a centrally contracted player.

Root also said he had spoken to his England team-mate about the incident and expressed his frustration at the "stupid mistakes" that are continually being made. "I've spoken to Brydon, and when I first found out my main concern was him as a person," Root added. "I don't think I'm speaking out of turn by saying he's very upset, very apologetic and understands his actions. I'm sure that when he gets his chance he will say that himself."

"But I don't think it's the isolated incident itself, it's the fact we keep making stupid mistakes away from the game, and it's detracting from what we're doing on the field."

In October, England white-ball captain Harry Brook was punched by a bouncer outside a nightclub in Wellington, New Zealand, with team-mates Jacob Bethell and Josh Tongue also present. During the Ashes tour of Australia, opener Ben Duckett was filmed apparently drunk on England's holiday in Noosa. And in June, former Test captain Ben Stokes and pace bowler Gus Atkinson broke a team curfew and were present when a member of England's security staff was punched by a Saracens rugby player in a London nightclub.

In this latest incident, Carse was with Durham team-mates Stokes and Matthew Potts, all of whom are under England central contracts. In Stokes' case, the England deal remains in place despite his international retirement in June. It makes a total of eight contracted players who have been involved in or present for late-night incidents in the past year – more than a quarter of the 30 men on England deals.

When he was reappointed as Test captain, Root confirmed he would remove the midnight curfew which was previously in place during Stokes and Brendon McCullum's regime after a string of off-field incidents during the Ashes over the winter. Root stood by that decision, saying he still did not think it was "the right way to go", but admitted it would take some time for the team to be trusted by the fans again.

"In many ways we're going to have to earn a lot of trust back, and it's something that we're going to have to live and breathe consistently," he said. "I might only be in this dressing room for a couple more years, but there is a group of players there who might be in there for another 10.

We've had a very strong discussion about what's expected of everyone when they play for England, what the standards are that we have to uphold, not just on the field but off it. "I've made it very clear how we want to conduct ourselves, and now we've got to live it and show people what I see day in, day out in the dressing room, which is a group of characters who know what it means to play for England and not having incidents define how we are seen as players. That's going to take some work and time."

Root also confirmed he had spoken to his predecessor Stokes, who was also seen in the social media video, but only to check in on Carse's wellbeing. He was asked what potential consequences Carse might face but said he would have no involvement as it is in the hands of the regulator and was keen to reiterate that he was focusing on moving forward and "building something to be proud of".

"I don't think the recent past has been a good look," Root, 35, said. "It's our responsibility to change that. I see the dressing room every day and how we operate around cricket, and it's very impressive. But at the same time, I can understand why we are viewed in a certain way and in a certain light at the moment. That's what lies ahead of us, to change that. I just think, if we don't put ourselves in certain situations, it's not something we will have to worry about."

If there was any question as to why England turned to Joe Root for a second stint as Test captain, if he is the right man for the job, then this address at Lord's removed all of those doubts. Root showed himself to be the leader England need in this moment.

Clearly ticking over what happened at the weekend, Root was able to strike the balance between letting his feelings be known, showing compassion for Brydon Carse, and not pre-judging the investigation that is ongoing.

There have been plenty of these kinds of news conferences over the past year: someone wearing an England badge explaining a misdemeanour, promising to do better. This time, when Root spoke, his words were plausible and believable.

Some wondered why Root was the one left to face the media, rather than members of the management team like director of cricket Rob Key or chief executive Richard Gould. Maybe it is because no one else was able to speak with Root's gravitas. No one in a suit, not Ben Stokes, nor Harry Brook.

Perhaps this is the most impressive display of leadership anyone connected with English cricket has shown for some time. It should leave no one – especially not the England players – in any doubt where their captain stands on this issue.

For any athlete to tell the world they are "hacked off" with a team-mate is a stark message. Root addressed his England team on Tuesday morning. If the captain was able to speak like this in public, one can only wonder about the sternness of the message delivered in private. Carse would not have been there, but plenty of others who have erred in the past year would have been in the room.

There will be no return for the midnight curfew, but Root was also quite clear about players avoiding situations that could lead to trouble. "Stupid mistakes," he called them. He hammered home the point that players will not find trouble if they do not put themselves in the places where trouble could find them.

But Root also regularly mentioned the notion of this team now having an opportunity to build something on and off the field. Put simply, their reputation in the eyes of the public cannot get much lower, so the only direction is improvement.

Amid all the controversy, English cricket is fortunate to have Root. Not only is he a bona fide all-time great, but also a superb example of how all sportspeople should go about their business on and off the field. Last week, before his first Test back in charge, Root asked his players to be "good role models and good human beings." They should look no further than their captain for inspiration.