They don't. But some Australians pronounce it like that.

You are wrong that people pronounce "rate" and "right" in the same way. "Rate" [reit] "Right [rait]

Speaking as an Aussie (who doesn't normally pronounce it like that) a schwa may slip in. Who pronounced it like that?

Perhaps the words are just said too quick for you. They do sound kinda alike.

at any rate is common,rate is common in everyday life,while right not that much,as an example the net rate,at any circumstance