I understand in the context of this manga, which is in Japan... where walking a bunch is a relative norm, the leg muscles are likely developed enough in the general populace to not have major issues.
Despite that, I still don't think jogging while (too) overweight is a good idea. Because all that additional weight is torture to your joints (even at late teens/early 20s). Yes you will build muscle in the process, but it's one injury away from sending you far back in motivation and progress. Which might also bury you in depression to boot. And that's a negative spiral.
Not that you can't do jogging when overweight, just in general lean toward strength more if you haven't been regularly active. So do bodyweight squats, planks/pushups, and something like pullups. Combination stuff that uses a bit of all muscles. Muscles burn more calories anyway, and a small increase can go a long way. Both for weight control and general stability in everyday life. And you do want a general balance everywhere because muscle imbalances can cause issues too.
For the bodyweight stuff, starting out at least, it won't take too much time too. Quick set of 5, 10, maybe 15 or 20 much later down the line once you're doing them with good form. Do a set when you use toilet, get a drink, while waiting for food to microwave. All adds up in the long run, and should be "easy" to do. And easy things are easier to make a habit out of.