Hi Desmo and welcome. I'm certainly no expert but I do have fig trees. Admittedly my climate is very different than yours but I have had a similar problem with figs not ripening.
My take on it is they need a certain length of warm weather time to ripen. Also plenty of water in warm weather when the fruit is sizing up will help matters. So in the UK if you have a long, hot (for you, I know it's all relative) summer you will very likely get ripe figs and if you have a cool, rainy summer, they may not ripen.
Probably extra pruning isn't going to make much difference except that some figs set fruit on last year's growth and others set it on new growth. It depends upon the variety. I'm not sure about Brown Turkey. My Celeste sets an early crop as soon as the tree is leafed out, then sets a second (main) crop on the new growth in the summer. I often get no ripening on the second crop, as the weather cools down abruptly in the fall here before they are good.
My inclination would be to leave it to grow for a couple of years, just pruning a little to keep it out of your way. Keep track of the amount of sun each season, and see what happens as the tree grows out from its recent severe pruning. If the leaves are healthy, and you have a long summer, you should get delicious figs.
