A lot of owners have made significant profits on their investments, certainly including DAL, NE and the NYG.
My calculations are that the total net worth is $45.69B (you can check my arithmetic).
http://www.forbes.com/nfl-valuations/list/
If the the annual profit is $1.39B, then owners are earning 3% on their investment. Few good businessmen would consider this return adequate. From a business perspective , they would sell the team, and expect more than 3% in other business ventures. Of course, I haven't added the increased value of the franchises. If they are say increasing at say 3% a year (a WAG), then the total returns are more reasonable.
All of this analysis has to do with NFL as a whole. Clearly, individual teams are more or less profitable. IMHO, there are clearly many teams that are not viable as a business. IMHO, as long as they are making some money, they can be justified as a hobby, even though the owners could likely earn more money elsewhere.