Yes, you can pick them and let them ripen on the counter. Picking them a bit early is a good way to transport them for sharing, too.

I have 3 peach trees. Contender which came grafted from a nursery, and two Indian Freestone peach trees. All 3 of my trees, I think are in their 4th year, this year. I had one peach last year -- just one on Contender. Here is a photo of one of the Indian Freestone trees from April 20. I grew the Indian Freestone trees from pits. By reputation they reproduce reliably, by seed. There is supposed to be enough genetic diversity that 2 such trees planted from seed near each other will cross-pollinate . . . Indian Free is one of the few peaches that requires a pollinator. The next one is Contender. The tree is more substantial as it was already grafted tree from a nursery planted in same year, but its flowers are not nearly as pretty. The fruit is good, though. I didn't see any honeybees this year. I see some fruit on the trees today less than the size of a pea, but some of the Indian Freestone buds are just dropping. I expect to get some fruit, just not nearly as many are setting as there were flowers. Less culling, but I'd rather cull fruit that are too plentiful than have sparse fruit.

It would be okay, however, if too much fruit is the problem then I would try thinning out the fruit.

Have you ever thin your peaches. I mean taking off 70% of all peachlets when they are young so the rest will size up and improve the fruit quality. anyway, I suspect that your peach tree is growing from a rootstock. The named variety was dead and the peach rootstock has taken over. That is why the peaches are small and poor in quality. this year the peaches have sized up could be because some of the flowered were killed by late frost or somehow young fruit drooped at a larher number than before leaving the remaining peaches to get up in size.