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. :)
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.
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.