Ah I have re-read the articles I linked to and, I think, I understand this better now.
What was troubling me was why protein causes insulin release when it doesn't raise blood glucose levels.
The second article, I posted, says it is because insulin also has a role in sending the amino acids from the protein to cells. However this insulin in the blood, when there is little blood glucose, could cause hypoglycaemia so the liver releases glucagon to counteract this. Other books I have read (eg Patrick Holford -Low GL Diet) say the release of glucagon promotes 'fat burning', Wiki talks about glucagon's role in using up glycogen stores.
So I think my conclusion is the downside of eating high protein is that it elevates insulin levels - which some researchers believe might have a detrimental effect, long term. (Although the 'fat burning' it promotes might counteract some of these effects.)
Taking this into consideration, I have decided to eat more of an adequate protein, low carbs (and only those which are 'natural' and unrefined) including (as unprocessed as possible fats and oils), with some intermittent fasting to boost weight loss.