Prunig is needed for sure. Google some images of commercial peach orchards and take a look, they'll show much less green vigorous growth. Quite a few guides for pruning available online too. You will see some talk of "scaffolds" and "fruiting wood" but these start to make sense when you understand the process. Ultimately you're going to want to open that tree up with quite a bit of pruning I suspect.

Peaches also generally benefit greatly from thinning and pruning of the baby fruit down to far fewer per limb. Many varieties are quite precocious and set more peaches than the tree could support. If you only have this one tree it won't be too big a job but it can be tedious. Again there is quite a bit of information available online about thinning the fruit.

Peaches will need spraying to produce good fruit, but it looks like that's been covered.

With some work over the winter and early spring you may just be posting about your first homegrown peach next summer!