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Fountain pens can improve your handwriting because they use nibs. Using a nib forces you to write at a proper angle and with proper form, so fountain pens can decrease pain from writing for long periods this way. Ballpoints and rollerballs don't use nibs, so better versions probably won't help you write better. Finer ballpoints, rollerball/gel pens can write more smoothly, and any decent fountain pen will write more smoothly and with less pressure than standard ballpoints, which may decrease pain in your arm. Fountain pens are probably the smoothest writing, however.
I think you touched on the main aspect as to why a fountain pen will induce better handwriting: pressure. Any cheap ballpoint pen has to be pressed onto the paper with a decent amount of force to get it to write. Same with roller balls, just not as much. A fountain pen, held at the correct angle with the nib aligned properly, requires basically no pressure to write.
If you need to press the pen to paper, you need to grip the pen tightly. Since your grip is so much more relaxed with a fountain pen, your handwriting will flow better. You can't have fine dexterity while tensing up all your muscles. Your hand will only flow if it's relaxed.