Getting to the East Midlands via Birmingham or Cambridge really doesn't make sense. If you drove that way, you'd be thought mad.A brand new HS line paralleling the M1 might as well be an extension of a Birmingham or Cambridge route unless you plan to serve Luton, Milton Keynes and Northampton.
The Birmingham route is defensible as an intermediate measure, as the line via Birmingham is being built anyway to serve Manchester and Scotland. With a huge speed improvement over existing lines, it's still faster despite going the long way around.
If you were building a direct line from London to the East Midlands, you wouldn't stop in the middle, whether for Luton, Milton Keynes, Northampton, Cambridge or Birmingham. Stopping high speed lines at every large town makes them pointless. If Cambridge needs a higher-speed line, it needs to justify that on its' own merits.
What might well make sense is if, once HS2 takes the load off the southern ECML, the Cambridge line is improved to allow 125mph (or even 140mph) operation with Azumas. But this is straying into hypothetical territory.