We still do social and company events. For many of the company events, HR handles planning and organizing. We do an annual company outing/team building and monthly lunch and learns. I do collaborate with other functions on the lunch and learns to organize the schedule.
For our last team outing, we went go-karting and placed people on random teams (people they wouldn't normally work with) and it made for a great team building event.
Social events (ping pong, corn hole games, after work happy hours) are really brought about naturally by employees in the company. I don't get too involved with organizing those things.
At my previous employer there was a dedicated social committee to handle company and social events. The committee was made up of multiple departments and the chair person happened to be the HR dept manager. We also had a young professional group that had a dedicated social coordinator for YP specific events like happy hours or group outings (hiking, biking, rafting, concerts, etc.) That company was about 1500 employees so the committee thing worked really well. The executives signed off on all events before the events were announced though so nothing crazy or contradicting to the culture happened if that makes sense.
At my current employer we just had a work party and the CFO organized it. I think it just depends who you have in your organization that has the time/interest. Those who actually have an interest in coordinating events are going to be much better at putting together something your team will enjoy. If you're worried from an HR perspective about the event just have final say go through the HR dept or the executive team.
We also have an activity committee. We do an annual picnic at a zoo or amusement park, a winter event at a waterpark hotel (we offer discounted tickets to all events), baseball game, and movie day (we rent out a movie theater).
We also have a community involvement committee whose main focus is adopt-a-family and local food banks and animal shelters. Each department adopts a family and does fundraising and events related to this. For instance, this week there is a sponsored motorcycle ride.
In addition, we have a health committee who plans softball tournaments, bowling, a race and other activities to encourage movement and health.