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If you figured it, just talk to the mother.
I've laid the reasoning out for you in spoilers below if you're stuck.
Without just giving you the answer, I'll lay out how I ruled out everything else to get the correct answer, assuming the puzzle doesn't have random elements.
Mom clearly states that no colour or pattern is repeated, so each must be unique. Thus with four options and four ages lined up, any colour that's listed explicitly as an older age must be incorrect.
The two types of shoes he mentions explicitly in his note to the shoe fairy are both from when he's older than five, so rule out the colour mentioned for each of these. The colour of boots given in the picture of Shay as a fireman at Age 8 can also be ruled out. You can find the correct pattern explicitly by looking at the bronze shoes, but not the colour or size. The medical records also mentioned that Shay grew three shoe sizes between Age 5 and Age 6, so if you assume the picture giving his shoe size is from Age 6, you then have all the information you need to full in the row for age 5.
No, the chart has a very specific point. It's the polar opposite of 'pointless'.
Point of the chart: to help you solve the puzzle, by showing you how you're meant to use deduction to get to the answer. Y'know, with your brain :)
It is still pointless since you cant use the chart in game. Though you might be correct pointless isnt the best way to describe it. Poor game design may be a better term. You can easily solve the puzzle without using the chart and in fact I did solve it without using the chart. While it is true I got the chat because I didnt solve it correctly the first time that was more a matter of lazyness. By the time I had got around to recieving the question I had long since forgot all the clues to the answer. It isnt a matter of using your brain so much as running back and forth over long distances to revisit clues you had no way of knowing you would need in advance or taking down note of every single thing that you see.
Looking at the drawing of Shay as a fireman, I thought the color of his boots were blue, so I eliminated that from the possible solutions. In fact the boots in the drawing were apparently purple though and blue ended up being the correct solution. The only reason I figured that out is because I accidentally misclicked on blue when I meant to click purple.
But the correct answer was purple! It couldn't have been blue because his mum said that: "she has never repeted a colour and a pattern" and the shoes in the picture were blue so his first shoes had to be purple then :)
From your answers I could guess that those colors were randomly done. In my game the fireman boots were red.