Heart rate zone issues

I'm otherwise very healthy, and not overweight, just unfit.

Regarding jogging/running. My heart rate goes into the 90% zone at a brisk walk (15 minute mile). Any faster than this requires picking up the feet and jogging, but then my heart rate skyrockets into truly dangerous territory. My legs fee great until the lack of oxygen sets in, and walking doesn't feel like it's doing anything to build endurance so I can eventually jog, then run.

My question is whether that fast walk is doing anything to increase vascularization in the muscles so, eventually, they can do more at a safe heart rate range. After a few months of this, it feels like the answer is no. But... what do you do to get around the heart muscle being the "bottleneck"?