I agree with the OP on multiple tab bars being problematic. What I think their designs are missing is grouping related tabs on the singular tab bar with dividers or colors.
One thing I hate about modern design, and this is on github etc, is the lack of use of color. Its all monochrome to look "professional" but color coding elements by function in a page can really make navigation so much easier. In the OP, just having blue, green, and yellow underlines with a gap under code / commits / branches, releases / issues / pull requests / projects, and wiki / insights / settings tabs would make visual navigation so much easier.
You could then border the file and commit list with blue and the statistics with yellow. The parts of the page related to each "topical heading" could be highlighted with color.