Hi Mate,
I think you already have a reasonable idea on what work's and what dosn't , target distance/ killing power and noise are all factors.
The only real reason to use a weapon of such limited power and range as the .22LR round is very close and silenced , as you already use.
In thick bush that has not been shot out , a silenced semi-auto .22LR will work well, but anything over 50 m is like pushing it. And if in the area you hunt distance is more variable than very short range 30-50 m , you well need a cartridge with more range and power.
In normal use I think the following cartridges all have good point's for hunting goats with, 22 Mag, 22 Hornet and the 223 Rem , 22 Mag being very cheap and good if you do not reload.
If you reload the Hornet and 223 Rem are very good choice's , and as you are a Kiwi , you can have any of these weapon's mounted with a silencer/suppressor , this well reduce the PR problem in curtain area's you hunt in.
I myself like the 223 Rem as a good goat round, but as you know already the humble goat is a very tuff animal , and shot placement is critical, I have seen goat's shot side on at close range with 12ga slug and show no reaction at all and run off into the scrub.
Sometimes the animal is dead , but it take's a while for the simple goat to realise it, I have hunted them a bit with a suppressed 308 rifle using full hunting load's , and the suppressor disguisied the sound of the shot enough so I had time to drop 3 or more animal's due to the reduced sound signature.