Actually the city's offer was to be 100% repaid, so yeah it is the Flames paying for the arena alone. That's like claiming you aren't paying for your house completely because you have a mortgage.
That's not true, though. The city's offer was to pay 33%. Not as a loan, but outright for the construction of the arena.
This is not a loan, and it's important to realize the difference. Property tax is not related to construction of the arena. It is a cost of owning land in a city. At the same time as the city receives money through property tax, the Flames make a massive profit by owning the building and collecting money from every event they host there. If you want to suggest that the city eventually will receive all their money back in that proposal, that is definitely true. And it is also true of the Flames, which by the Flames' logic means that no one is paying for the arena. Or everyone is. Hard to say.
Regardless, one thing that the Flames' and Bettman's whining makes clear is that the Flames have a great deal of money to make in a new arena being built. They are apparently losing money hand over fist by not having a new arena. They want the city to contribute more because Calgary's citizens benefit. They get to watch the Flames in a better arena, and they get all these concerts they're currently missing out on. But they benefit by going to those events at the new building. For which they buy a ticket. On which the Flames owners make a great deal of money, and far greater because of the new arena and all of its glorious amenities. Thus the citizens will pay for that privilege just like they pay for the privilege of having any kind of business in town. There's no reason to do mathematical gymnastics to dodge that.
Imagine a circus coming to town and suggesting that the city should fund some of their setup costs because the citizens will now have the privilege to buy tickets and go see it. Doesn't make any sense.
The Flames have made it clear that they're going to make out like gangbusters by having a new arena compared to the old one, so the issue clearly is clearly not in whether it will improve their bottom line, but by how much. For the city, this will be a net loss compared to having a normal business in that space paying for their own premises like normal and paying the same property tax like normal. But it's one they're willing to take because of loyalty to the team.