I agree with this perspective to a point and John Oliver did a solid episode on publicly funded arenas. I've heard arguments on the other side that also sway. I've seen that the arena is estimated to cost $627M and could go higher. Even though Ilitch is personally worth 5 billion+, I doubt his company has that amount of free cash to spend at one time so I understand if a certain amount coming from government funding.
One thing to consider on the naming issue, is that the Ilitches have essentially turned away a huge amount of money by not selling the naming rights to another company. Maybe it's their interest to have the advertising (although I don't think it's that much extra exposure - we all knew the new arena would be lined with Little Ceasars venders, signage, and they would give free pizzas at the games just like they do at the Joe). But think of the alternative of them taking a huge amount of public money and then using the arena to bring in an extra money to cover part of their end (Blues got 70 million for 20 years naming rights for the Savvis center for a random example) Probably everyone would prefer Howe arena or honoring some local hero, but I think Little Ceasars is the best option for corporate naming.
I get WHY public funding is often necessary, I'm not even against public funding. I just think that if you're going to accept public funding for your teams new arena you should probably make an effort to represent and validate their contribution rather than be wholly greedy and promote your brand to their exclusion.