No that would be normal. That charge voltage is just fine. Measure battery voltage after a couple hours of sitting dormant, or else run the headlights a couple minutes with engine shut off and see what that gets you. What you have now does not seem too bad A healthy fully charged battery is about 12.6 but this varies a little and with temperature
Another way to get an idea of battery "health" is to disable the ignition, and crank the starter for about 15 seconds while monitoring the battery voltage. The higher the better, but if it's above 10.5 consider it "GOOD"
One thing I've recommended as a "work around" for the ignition / VR "run" line to "fix" that is to cut the "run" (normally dark blue) where it comes out of the bulkhead. Use the bulkhead side to trigger a (such as) Bosch relay, and connect the engine bay end of the blue to the switched relay contacts. Feed the relay with a fuse/ breaker off the start relay "big stud." On a side note you can run the relay coil ground connection in through an unobtrusive switch to ground and create a simple anti-theft device.