Sheesh man, don't choose your frets in order to have no string trees! That's ridiculous. Fret size is really a personal thing, you should get the size that you like the best. Besides, the difference between the smallest and highest fret is a tiny amount that will not end up affecting the break angle behind the nut significantly.
\I'm 100% sure from your description that your issue with string trees is caused by either 1) a bad string tree, which you should replace at a cost of $3 and five minutes of work (replace with a barrel or roller type, maybe put a little bit of graphite under it just to be extra sure) OR 2) your nut is cut too tight, which is causing all of your tuning issues and the pinging you hear. Just take your strings off of the tree and play for a while to make sure it's the tree and not the nut. String trees may or may not be necessary, which is what this thread turned into, but they are definitely not the major problem that you seem to believe.
Nearly all tuning problems are caused by the nut. Strings should lie at the bottom of a groove in the nut, being held in there by downward pressure from behind the nut, exactly the same as happens on the bridge. They should not be held in place by the sides of the nut slot - that causes tuning issues. String trees just ensure that the break angle is sufficient to get that downward force - without a string tree, you might or might not have enough angle to need it. An angled peghead guarantees plenty of break angle, which is why all acoustics and Gibsons, as well as banjos, mandolins, violins, etc. etc. , has an angled peghead. Fender's straight head was an economy measure, and the string tree is a cheap and dirty way to fix an engineering problem that resulted.
In other words, stubhead is right.