Yeah, drawing animations frame-by-frame makes all the difference.
One reason why modern 2D games animations' may look awful is because the character is usually made out of individual pieces, and animated using 3D techniques (i.e. interpolation).
They end up looking like those paper dolls that have the limbs attached to the torso with a nail.
Something like this
For games with a "puppet theatre" aesthetic it might look good, but for anything else it might look cheap.
Vanillaware games use that techinique, but they've managed to hide it pretty well. The "puppet aesthetic" is still noticeable though.
There's an old interview with the developers of Wonder Project J2 if you're interested
Wonder Project J2 – 1995 Developer Interview These two Dengeki SFC interviews cover the making of Wonder Project J2, the sequel to Almanac / Givro & Enix’s hit “communication adventure” game and one of only two Enix-published games published for the Nintendo 64. Producer Hiroki Fujimoto and...