I do indoor biking, for this exact reason. But, it has a much lower impact on my health. Maybe it's because indoor bike don't move, thus not involving as many muscle groups as real bike.
The efforts are not distributed the same way. Obviously biking will exercise your thighs but not much else. While running uses abs, all your legs muscles groups. I also believe that we're wired to be efficient at running and thus running tickles your body in better ways. Also there's a nice feeling when your center of gravity lifts up.
Even if I can go fast on a bike for 1 min, I can barely sprint that long and surely slowly and I end up exhausted differently. In the end I prefered what running brought me. It may be very subjective though, even if I enjoy all sports, I always had a thing for running.
This is all crude opinion, I'd love if anyone with bio/biomech knowledge could shed light on the subject.
Cycling is more fun than running for me but exactly because of all the reasons you mentioned I prefer running over cycling. What I also realized is that nothing works as good as running (at least for me) for weight loss if you also change your eating habits (you need to have a calorie deficit in order to lose fat).
For around 9 weeks I managed to go running for 10km every second day and then my two weeks vacation happened. I'm trying to get back in again because I'm extremely frustrated on days I don't go running. I cannot emphasize enough on the stress relieving part. For me it's around 50 minutes of complete peace of mind where the only things I focus are breathing and moving my body.
One thing I recommend is to track your runs because it's fun and interesting to see your improvement over time. Bonus points if you track your heart rate as well. :)
I use some android app (fitness run or something like this) to have logs. Google fit is nice even though it's not precise GPS wise (relies on Google Maps trip logs).
A smartphone is a bit annoying to carry and use, I might invest in a gps wrist so I can pause / resume with a button.
Right now I'm at 2x2.5km jogging + 5km of walk. And even though I don't have a calorie deficit, I already lost significant amount of fat and got back muscle mass (the bliss of sore muscles right? ;) I'm sure even with a normal diet it's a lot better to have a more active metabolysm as it is now.
Good point for the muscle groups involved. Maybe you could bike for cardiovascular health, and for muscle training add some squats (a simple complete workout would be push ups, pull ups and squats, pull ups require a bar though but it's a very useful investment for your back in the long term)
Cardio is the sole reason I do indoor biking. I was very sick and couldn't do much, so I tried having mild biking effort as a way to rehabilitate my heart.
I used to do exactly what you suggest: bike + full workout. I liked it a lot, but it's not integrated as in a sport. It was odd for instance, I had much more "power" than ever, so much that I couldn't run anymore because my thighs were too strong and I had to relearn how to land my foot otherwise I'd miss the ground.
The efforts are not distributed the same way. Obviously biking will exercise your thighs but not much else. While running uses abs, all your legs muscles groups. I also believe that we're wired to be efficient at running and thus running tickles your body in better ways. Also there's a nice feeling when your center of gravity lifts up.
Even if I can go fast on a bike for 1 min, I can barely sprint that long and surely slowly and I end up exhausted differently. In the end I prefered what running brought me. It may be very subjective though, even if I enjoy all sports, I always had a thing for running.
This is all crude opinion, I'd love if anyone with bio/biomech knowledge could shed light on the subject.