There are something like 15 judges in CA5 (5th District Court of Appeals). 3 of them heard this case and gave us the ruling we had. Elliott can appeal to try and have the case heard by all 15 judges.
To call it a long shot would be generous. en banc appeals are very, very rarely heard.
Technically the NFL did not file early. They filed in the NY District after the abritrator's ruling was issued.
The problem with lawyers - and probably the main reason I gave up my studies and went in a different direction - is that the nature of the profession is such that they profit tremendously from conflict. 99.9% of them are more interested in their *own* best interests and not their client's (and they are not even remotely worried about society's best interest).
I had a good friend who was getting divorced. He and his then-wife agreed on all property matters except the house. He said he would move out and let her keep the house (and all equity in it) for $25,000. She saw a lawyer who advised her not to accept that offer. My friend also suggested mediation but, on the advice of counsel, she refused. They went to court 6 times to resolve the matter. In total, she spent around $35,000 on lawyers' fees and still had to give my friend $12,500 to get him out. (My friend was smart; he represented himself in court so even though he kept losing, it didn't impact him financially as the total gain would not have justified the expense)
His wife's lawyer was the textbook example of a small, 1-person law office deciding she needed billable hours more than she needed to represent her client's best interest. An ethical lawyer would have said "you have a strong case to get the $25,000 reduced, but the expense of going to court will simply not be worth it."