That also makes it more clear that while reaching 0 (i.e. infinite Beta) is impossible, negative temperatures are possible (this occurs in processes that undergo a state inversion like in a laser / maser). [2]
Here is a plot of Temperature vs. Coldness from 0-100 C, scaled such that water freezes at 100 coldness units and boils at 0 coldness units[1]: https://i.imgur.com/fluOSY2.png
It traces out a pretty gentle curve, which seems like it's not appreciably less linear than our ability to perceive differences in temperatures (from eyeballing this: https://biology.stackexchange.com/a/28522), so probably for day-to-day use inverse temperature would be a perfectly acceptable substitute.
Of course, this is just to say that if we were starting de novo today in determining how to discuss temperature, coldness could be a reasonable choice. I don't think it actually offers substantial advantages over the current system and it would likely be very costly to switch everyone over just for the sake of making it a bit easier to think about why temperature has an absolute zero.
[1] Python source code: https://bpaste.net/show/76bf4f7c36cb