Nami also doesn't stop while casting Tidecaller's Blessing. Likewise with Janna's Q, Thresh's lantern, Ahri's W and countless other spells, including most AA-modifiers, which a lot of happen to be AA-resets as well.
I never mentioned "skillshot" champs in any specificity, but I don't believe that has any relation. Look at Ezreal and TF, they are BOTH "skillshot" champs and both of them received compensation buffs when Lich Bane was nerfed. Using Sona's "100% accuracy"(which it is not) is nothing more than an excuse. Every AA in the game has 100% accuracy(except Kalista) and countless spells do as well. Including: Nami's W, Annie's Q and Janna's W.
As a matter of fact, Sona's spells are NOT "point & click," "100% accuracy," "single target" or anything of the sort. It's auto-target, which is less reliable and harder to use. First, you need have your range perfectly memorized, just a slight miscalculation results in burning the spell, wasting a cooldown and mana. Even if you are in range, you have no control over its auto-tageting system, the only way to guarantee it working exactly the way you want is to position yourself to be perfectly in range of ONLY your selected target(s) and NOBODY ELSE.
Many champs can cast their full combo instantly. Annie does it by bashing your face against the keyboard. Watch Froggen the patch immediately before Ahri was gutted and you'll see her do it as well. Even Anivia is literally just pressing E and R at the same time. This isn't something that's either new or unique.
Lich Bane for most champs is seen as a "win more" item. Any champ with good scalings prefers the extra 20-40 AP from other 1st-item purchases(Echo in particular is ALSO an artificial scaling, but w/o sacrificing AP). If you need Morello or Grail, that 100/120 AP item becomes a 2nd-purchase. You're gonna want Deathcap at some point and you eventually need Void Staff guaranteed. Boots are nigh-close to mandatory and you probably want a defensive item or 2. While it's true Lich Bane greatly amplifies your damage, it's difficult to find room for it in your 6-item build unless you completely forgo buying a defensive item entirely. Most champs have high enough scalings to get the job done without needing to buy a sub-optimal AP item for 1 more artificial scaling.
The champs you typically see buying Lich Bane tend to have poor AP scalings(often due to their Spellblade reliance) and usually don't suffer from mana problems. They NEED the extra scaling on their kit, but in turn this also means sacrificing the 20-40 AP doesn't hurt them, this includes Sona. I'd compare it to a self-fulfilling prophecy in a way: they have to buy Lich Bane because their scalings are low; they're allowed to buy Lich Bane because their low scaling lessens the blow of lower AP. Rito is LITERALLY balancing them around the fact they commonly buy Lich Bane. Perhaps if their scalings were buffed, even they too would stop buying it for its low AP stat that disfavors it as an early item, but that's only conjecture.