I have two persimmons, two asian pears, a cherry tree, blackberries and raspberries and asparagus among my semi-permanent crop in my suburban lawn.
Here's why I don't think you should plant from seed:
1. The varieties that are best for large-scale cultivation (even locally-grown) are not necessarily the same breeds that best suited for a small-scale yard. Farmed orchards need fruits that will look best, travel best, store best and are generally kept on a spray schedule for chemicals (even organic fruit is subject to a rigorous pesticide schedule). But guess what? In your yard, you're probably not going to keep a rigorous spray schedule and you don't need long-term storage. What you want is disease resistance and taste (with pest resistance being third).
2. There may be many kinds of fruits that are extremely well suited and care-free to your yard, but that aren't grown commercially for a variety of reasons (the most important being that they're not economically viable).
3. It will take years for your fruit tree to establish and grow fruit. If you buy a tree, you may have fruit the very next year. If you plant from a seed, you may be waiting 6-7 years. Sure, you may have paid $50-$100 for the tree, but it's probably worth 10-15 dollars per year to speed up your timeline. Certainly, the joy of picking your own fruit from your own tree is not something you should delay!
4. Grafting issues. Even though your fruit is locally grown, that doesn't mean the seed matches the rootstock. You aren't necessarily planting a copy of the same tree that your fruit grew on and the results may be dissapointing. Your dissapointments will come in the form of dead or unhealthy trees - not the end of the world, but it only delays your success further.
I've cheaped out before - never so cheap as planting seeds - but cheap enough that I've spent under $10 on the equivalent of a stick. And five years in, my stick still has no fruit, but the trees I spent $35 each on bust out as much fruit as I can eat every year since the first year after transplant.
The internet tells me that if they sprout and are healthy, I should have fruit in 3 years and a twenty foot tall tree in about 5.
The internet is wrong. This won't happen - at least not in tangent. A cherry might get to fruiting age in 3 years, but then it won't be twenty foot tall. I've never really seen any tree get to twenty foot tall in 5 years - maybe maples and willows. My fruiting trees that are 7+ years old fruit very well, but aren't even close to 20 foot - my tallest is probably the asian pears, at 12-15 feet, and they are tall and skinny, not shade trees, by any means.