There's a finite list of approved arbitrators (about 60 I think) and a subset of those that were best qualified to adjudicate in our particular case (let's say 20). So the panel could only come from that 20-person pool. And two of those had already been picked, one by us and the other by UEFA.
The process was that someone had to suggest a name, with the other party having the right to veto that choice. Had UEFA vetoed our choice of Rui Botica Santos, and had both parties failed to agree on another choice, then CAS would have selected a chair.
Its like going out for a meal with your mates or wife.
You: "Do you fancy Indian, Italian or Chinese?"
Wife: "I fancy a Chinese."
You "I'm fine with that".
Then afterwards you complain that your arm was twisted, you didn't want Chinese and you'd rather have gone for a pizza.