Just to add some experience, just had an S5 in with similar issue, apparently its wear in the clutch release bearing / or (throwout bearing if you are American), and the Audi dealer fix is a replacement clutch, which is major expense. Fine if it's warranty or goodwill but unless the clutch is faulty in other ways sticking slipping very stiff etc there is a simple fix to the starting issues. Because of the wear in the release bearing the slave cylinder is running out of travel before the clutch pedal is getting to the end stop so not trigging the engine start switch. .
The switch has other functions and is a sealed unit, so no physical moving parts on the outside so is likely magnetically trigged by something inside the master cylinder. Anyway the fix is simply to lengthen the slave cylinder push rod so it has more piston travel. As a test I just used a small nut between the rod and the piston but later just cut and lengthened the rod by 5mm. car now starts with pedal about half inch from fully down, rather than having to press it through the bulk head like before to get it to start.
Remember this is a get it useable fix and at some future point the clutch, (or at very least release bearing), will require replacement at kidney selling cost, so if you can get the proper fix and or it's under warranty don't mess with it but get Audi to do it free..