If you grow a Lychee to fruition from seed, it will likely take a long long time and the fruit can taste horrible with huge seeds and little flesh. Imagine planting just one seed, waiting 10+ years for Fruit only to find out the Fruit sucks. In that time, you would have wasted the space in your yard and a Lychee tree will probably be fairly large by the time it fruits.
The various Lychee selections have been propagated for a reason. They have been selected for particular flavors, color, size of seed, amount of flesh, ripening time, dry type flesh, water type flesh, etc...
I grow Lychees from seed but I am only using them as rootstocks. I grow them up to a certain size and innarch them with a named variety. I’m doing this because like you, I want a Lychee tree with a strong taproot.
I would highly recommend purchasing a named variety of Lychee so you can ensure you get good quality fruit.
Simon