DH and I were ‘sporty’ when we met, and continued our activities after marriage. The only sport I insisted the kids (2 sons) do was swimming lessons - because it could save their life, and possibly become a decent job for a youngster. Besides lessons, DH and I took them to the local pool for the family swim every weekend. Both became lifeguards and swim instructors at 16.
We did sports as a family, but it was more about leading an active lifestyle. Some soccer (DH and I coached) and cross country skiing when they were little, downhill skiing/snowboarding when they were older.
My kids didn’t do sports through school other than cross country running and track and field, and only in elementary and jr. high - not at the high school level. (They did do band at school and with a community youth band, but that is another story!) One is introverted, super tall and thin, and the other husky, extroverted, and has a vision impairment. They saw DH and me do our thing, and they came to my races as spectators, and sometimes did the kids short races. DS#1 couldn’t wait to get into triathlon (you had to be 15 back then - no kids events) we went ‘halfsies’ on a racing bike when he was 16 and he joined a local cycling club and LOVED it. Once he got started we travelled to races together, and now triathlon is his hobby sport. DS#2 found his sport around 15-16, when some friends got into martial arts. His father and I weren’t keen about his choice, but grappling sports (Jiujitsu, judo, and wrestling) became a passion and he even volunteered at the Commonwealth Games at the judo venue a few years back. He weight trains several days a week to build strength for his sport, and has also taken courses on power lifting. He also got into rowing in university!
I will admit that the boys followed more in my footsteps - but that was because I dragged them around to all my activities, and was prepared to accommodate their pace at the expense of my workout, so that they would have fun. DH didn’t do that so much, so they didn’t get the same exposure to his sports.
I agree with FI - that the surest way to turn your kid off a sport, is to force the issue and 13 is a tricky time. If you want him to have more physical activity in his life, assign him the chore of snow removal this winter, and don’t drive him everywhere - get him to walk or bike. He will likely find his passion over the next 2-3 years, and it might surprise you!