I think you are right about last night. Oddly I thought that was his worst game for us. In the other games he has been very assured defensively whereas in others he looked very composed and strangely bigger and more physical.
There's no doubting his quality going forward though and the quality of delivery into the box players like Lukaku will thrive on.
I am inclined to agree with you on whether he signs or not. If he can play 1/2 dozen games between now and the end of the season that is a decent number. I hope by playing him Martinez has reassured him that there will be opportunities going forward for progression into the first team. He only needs to look at Stones and Barkley for that as well as Deulofeu before him that the manager will give younger players an opportunity, and do it in a manner that they improve steadily. In all honesty he would be silly to move on to a manager like a Pulis/Allardyce who would ditch him at the first sign of an experienced Carthorse becoming available.
As you say though it's up to him. We don't know whether he wants more money or more game time, and that can only be solved by offering more game time. I think under Martinez's guise the lad could go on to play for England. I don't think Shaw is any better than him. But he does need to meet Martinez half way and recognise it may be a couple of years before he is a regular first team player.
He reminds me of John Stones when he first came in. You could see Stones had real ability, outstanding attributes in areas. But he was a bit shakey when he first started and would get caught under high balls quite a lot. Stones has really improved defensively though over the last 12-18 months. This should show us two things. 1) Garbutt if he has his head screwed on should want to remain with Martinez to continue to improve. And 2) Martinez can develop and improve defenders.
Stones was lucky as he had Distin in front of him who has thrown his toys out of the pram. Baines won't do that, so Garbutt will need to wait a bit more. But as you have said, he needs to see this for himself.