Bloomberg let you do philanthropy work during working hours, and they assign you some amount of money depending on the hours you do that you can donate to charity.

I met a few people during such events, which is more or less what you say: you are with people from the company (not necessarily teammates, we did it as a team only once) and you might be helping public parks clean, or take trash from canal on a canoe, or sort food in a food shelter. It's a nice way to take a break, meet people from other areas, feel good about something you do, and managers have been very supportive of that, and they do it too.

I think that is a good balance of what you are talking about