Depends entirely on the issue man...lol. If you're a technician getting assigned tickets, yes you're going to be leaving your desk and working on the issue physically (unless it's something that you can just do remotely, as that's also an option...which doesn't matter if you have a ticketing system in place or not...). If you're a company that has 'separate' entities for desktop support, networking, security, etc., then yes you would just pass the info along/assign the ticket to the appropriate group.
Otherwise, if you're a tech that's going to be going out and actually working on issues...being assigned a ticket to work on (or creating one) shouldn't be an issue. You're still going to be in contact with the person that contacted you for help, still can chit-chat or w/e while you work on the issue, etc. None of that would change - it's just that you actually have a good way to keep track of who has what tasks assigned to them to work on.
You're lumping 'everything' together and acting like a ticket system will somehow put up a barrier and make people not be social with the client - I fail to understand how you think that will happen.
You can still do exactly what you're doing, but have a better way to keep track of past/ongoing issues.