There's no requirement for arbitration to find some middle ground between the parties. Are people confusing it with mediation? It's perfectly common for an arbitral tribunal to come down fully on the side of one party.
Arbitration is simply a form of alternative dispute resolution that avoids litigation through state courts by having a dispute examined by a neutral party or tribunal. In general, an arbitration decision is usually binding with no right of appeal, and the procedure is usually streamlined as compared with litigation. As a result, matters tend to be resolved more quickly and at lower cost, so it's a popular choice among parties to contracts.
In City's case at CAS, I think that a big issue was that UEFA's choice as arbitrator was Prof. Ulrich Haas. He's a German academic but has his own law practice in Zurich as well, and he happens to be a regular choice of UEFA as arbitrator in the many cases they fight before CAS. I'm told by specialist practitioners that, in most arbitration cases, arbitrators aren't especially tied to the party that nominates them as it could be an age before that party is back before the same tribunal. Not so here, obviously.
I'd suggest, therefore, that Prof. Haas is an exception to the above general rule. I suspect he knows full well which side his bread's buttered and didn't want to risk his cosy little arrangement with UEFA by voting against them. IIRC, didn't his trainee (or equivalent) subsequently produce a jaw-droppingly inept, but very revealing, analysis of the case moaning that we'd got away with it?
As for the chairman of the panel, the rules stated that he'd be chosen by agreement of the parties or, failing such agreement, by the CAS itself. We put forward a name for the third, i.e. 'neutral', arbitrator and UEFA consented. Idiots such as Tony Evans - who has zero informed perspective on this but was mouthpiece of our enemies - later cited that fact as showing UEFA failed to put their full effort into their defence before the CAS.
Finally, I find myself in agreement with the CAS that we had to be fined for non-cooperation with UEFA. I understand why we took that stance, and indeed supported it at the time, but it would have make a nonsense of UEFA's investigatory powers in future cases if we'd not been sanctioned. Nobody would ever cooperate with them again, even where UEFA was demonstrably correct and not conducting the kind of lamentable kangaroo court that we saw in the case of MCFC.