The analogy to a civil case is closer than to a criminal case, but inexact.
Civil cases can result in injunctions against certain behavior, so that's a loss of liberty in a sense. But not loss of liberty to do one's job.
In a civil case, there are interrogatories and motions for discovery that require the other side to produce the facts within its possession. There are various grounds for objection, but you can't just say "No, I don't think I'll give you that." So Brady was better off on this point than he'd be in court.
On the other hand, in a criminal case, the defendant enjoys both the higher "beyond a reasonable doubt" standard (which the findings here would obviously never meet) and a right against self-incrimination.
The standards here are not, of course, actual court standards. Essentially, the NFL's power to impose sanctions is a contractual power. In an appeal, the team, the Union, the player and the league must find their justification in the contracts that bind them all together, elucidated to an extent by past instances that they all lived with as resolutions.