This always seems to be the answer, but...
Talk to the player first.
I'm assuming you have some means of contacting your players outside your normal game time, if only to set up game or let each other know of cancellations or emergencies. Send your player a message, something along the lines of "Hey, I've noticed that you seem dissatisfied at game lately. Is there something going on you're not happy with? I want to make sure our game is fun for all of us, so please let me know if you have any concerns."
See how he responds to that. If he responds with a list of concerns, use those as a starting point. If they're things you can address, try to do so. If you think his problems are irreconcilable with your game and/or the other players, then say that, politely, and suggest that he might want to look for a different game to join.
If his response is neutral or brushes you off (such as "nothing's wrong, see you next game"), then you can let it go for a session or two. See how he behaves, and whether he raises the issue himself.
Ask another player to initiate communication
If you're not comfortable approaching the problem player directly (or don't have the means to do so), you can speak again with the person who told you this player wants to leave. It sounds like the go-between is friendly with him, so you can ask the go-between to, next time the player brings up the issue of wanting to leave the campaign, tell him to talk to you about it.
Don't use the go-between to actually convey messages like "I heard you want to leave, what did I do wrong?". That kind of thing usually gets lost in communication, and it puts the second player in an awkward spot. Just ask him to pass on that you're open to hearing your players' concerns and that the player is welcome to talk to you if he has a problem.
Don't jump the gun
Either way, don't rely on hearsay to drop a character and his player from the party. Don't allow the other players to sacrifice this guy's character unless you have his explicit buy-in (or if it comes up naturally in-game and the player himself is all for it). Don't assume that he meant what he said about wanting to drop, either - I've had players complain about my game in moments of frustration, but when I ask them if there's something I can do better, they reassure me they're having fun and were just briefly frustrated.
TL;DR: Communication is your friend!
Talk to the player. Follow his lead, and don't make assumptions based on what other people are telling you. If you handle this issue with grace and good will, then it's highly unlikely the other players will see a reason to stop playing a game they enjoy just because their friend did.
In response to your sectionals:
Do NOT repeat things for them. If they miss it, they miss it, they have earned no special treatment - do not offer them any.
As much as I personally dislike it, a time limit (say 1 min) for each player, each turn may be in order. (the time limit is only on committing to an action, not actually finishing it ... ie large meteor swarm could take a while to resolve)
- Losing a sense of cohesion:
If their characters are no longer fulfilling a prerequisite for something, then their characters should suffer some penalty or issue. This is an item for the DM to discuss with them. Although it could lead to booting the characters (not players) from the party, it could lead to new character creation .. or even player dismissal ... although this isn't necessarily what I'm leading at here ;)
If the players themselves are not abiding by the rules of the table, and communicating with them has failed, I'm not sure I see any other option than to start enforcing stricter penalties on them, or asking them outright to leave. I know you didn't want to hear that, however, some people just don't "get it". shrug
You don't need to handle this behaviour, neither does the GM. Just simply point out . "Hey, it was your choice ... live with YOUR choices". These hard heads need to grow up .. short and simple. :)
Best Answer
As with any motivational approaches, there's the carrot and the stick. You have to be careful to not simply be permissive of the late behavior, or else you won't incentivize the people who are showing up on time to do so.
Start at a known time and allow a buffer. On our group we have a "doors open" time and a "game" time, to allow for people to show up and shoot the bull. (Or for weekend afternoon games, we'll meet at noon for lunch, 1:00 for play). So in your case you might do doors open at 7 and game starts at 7:15. This provides some incentive of fun free discussion time to get there earlier and set up. But start on time, and if they are late they miss a little and get some peer pressure from the group for interrupting.
Bennies. Give something cool out at the beginning of the session. Whatever your game system allows, a FATE point, an action point, a healing surge, whatever. It's only for those there at 7:15 sharp. This makes the point and provides a positive incentive for on time gamers.
Timing. Try to not kick into a major combat in the first ten minutes. We have two chronically late players in our Jade Regent game and we find ourselves dragging our feet as players when there's an early combat coming because we don't want to be under strength. With on time players it's cool to stop right outside the "big double doors" to pick up the boss fight next time; if you have chronically late players you may want to stop a couple "rooms early" so to speak.
Talk to them. It's possible that they have work or other stuff (traffic?) that just makes it impossible for them to hit 7:15 sharp every time. The game is secondary to the rest of people's lives and you have to understand that. But you can also express that if there's not something competing, but just general lateness, that it affects you and the other players and you'd like them to not do that.
Penalties. Don't do it. You may be tempted to do the reverse of Bennies and penalize latecomers. It will create hard feelings. Sure they're "being inconsiderate" but this isn't a business venture, it's a game among friends. I'm all about people acting right but the second you start putting the game above people you've moved into antisocial personality disorder territory. Especially when we're talking about "oh Lord they're 10 minutes late" - that's very low on the spectrum of disruptive behaviors.