Do higher tier Anipods have better catch rates?

Tier sets a base rate, which means the pod you throw decides your floor before anything else about the encounter comes into it. Throw a low tier pod at something strong and you should expect a lot of failed attempts. Higher tiers raise that base rate and are what make stronger creatures realistically catchable, so they are worth hoarding rather than spending on whatever wanders past you on the road.

That gives you a simple rule for managing stock. Commons and anything you are catching in bulk go in the cheap tiers. Your top tier pods are for the catches you would be annoyed to lose: high potential wild spawns, and the Aniimo that turn up in alternative colour variations in the open world. Wild spawn rates for 9/9 and 10/10 potential Aniimo have been increased, so you will run into more encounters that justify a good pod, not fewer. Budget accordingly.

Type matters as well as tier

Tier is only half of it. The Tumbler gives increased capture success in specific situations and reaches a 6x multiplier during gather events, which is a bigger swing than moving up a tier normally buys you. In that window the Tumbler is the throw, whatever else is in your bag.

The flip side is that the bonus is tied to those situations. Outside them you are back to comparing base rates, and the highest tier pod you own is the safest thing to throw. So carry both: the tiers for general reliability, and the Tumbler for the moments that pay it off.

Not settled: The 6x figure applies to gather events specifically. The other situations where the Tumbler improves your odds do not come with a stated multiplier, so you cannot yet work out when it beats simply throwing your best tier.