All you should need to do is to set up the log consumer in your script. eg.

use strict;
use warnings;
use Mastodon::Client;
use Log::Any::Adapter ('Stdout', log_level => 'trace');
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This should set up logging to STDOUT (since you asked for "get log output onto my screen") with level "trace" so that your call to is_trace will return true.

While I do use Log::Any, I do not use Mastodon::Client so if that module is doing something quirky then you may need to dig a bit deeper.

I suspect that if I knew more about Moo and Roles this would be more obvious to me?

Those are unrelated concepts, so there's no need to worry about them just to get Log::Any to trace your code.