This would be a silly restriction to expect people to run to catch a gate, considering that airlines do not have a policy of carrying only those that are fit to run.
They have families with small children, elderly passengers, passengers with disabilities, passengers that are new to the airport, very young passengers, passengers with language issues - and yes, even those that are slow walkers.
That's why you see in airports people (airline staff) running around trying to chase people that are not at the gate and their flight is about to depart; and similarly why you have golf carts zooming about carting people from one gate to the next; and you have travelators to help with the flow of traffic.
I have seen this all the time, when there is a tight connection usually at the arriving gate you'll have airport staff ready to escort people directly to the gate (sometimes, they even escort passengers through priority queues - for example, for immigration/security) all in the name of not missing the flight.
I once had to run ... well walk briskly ... to catch my connecting flight because I did not want to waste my time at the airport and needed to be home. The connecting flight was two terminals away. It was a very stressful walk.
You are only wasting your time if you stroll along knowing that you will miss the next flight. They will hold the gate for you as long as they can, so you'd have a high chance of catching the plane than missing it.
If you were to miss it, as the airline is obligated to carry you to your final destination - they will most likely offload your bags (if any) and then offload you from the flight. Once you show up - they will have to reschedule you on the next flight.
Holding a flight (delaying it) costs the airline dearly so they want to avoid doing it.