There is a middle ground to being a breathless doggy paddler and Michael Phelps.
Swimming lessons from the right teacher are the key. I can swim really well, because I was taught by a good teacher, but I couldn't teach. That's why I would send a child to swimming lessons, probably from about 5. I don't think it's a good idea to teach your child to swim unless you are (a) a good swimmer and (b) a good teacher.
People think one-to-one tuition would be great but at that age a small group of about 5 children is much better. I'd say that for adult novices too. Bonding in a small group helps learning and makes friendships. That, for me is the point of swimming if we are going to use it as a recreational activity.
I don't agree that swimming is a life skill and never will. But I do agree with various PPs who've said it is a very useful thing.
My parents sent me for a number of reasons: they wanted me to learn to swim so I wouldn't drown (I think that's wrong. I think people who can swim are the most likely people to drown because they have a false sense of security. That's just my belief. I can't back it up with data so don't bother asking); they wanted me to make friends - I did; they wanted me to get a sense of achievement from a physical activity - they'd tried gymnastics and ballet but I was rubbish. I was very good at swimming.
For fitness, it's not as cheap as running but it's close if you go to municipal pools. If you are shy about your body you only have to brave the walk from changing room to pool. No one can see you in the water.
Because I was taught technique I can easily put on the cheapest black school-style Speedo and do fast lengths for 45 minutes and then get out and get on with my life. Perfect