I was playing Kayle on the PBE today and I couldn't help but notice the damage coming off of her ultimate. And it made me think "Wow, this burst is really nutty. Does she need this?"
Uh, yea. She does. The change was made purely to be able to give some counterplay for cases in lane or 1v1s where Kayle would ult herself for a free 2-3 something seconds of invulnerability and time to melt you down with her autos. In exchange for not being able to auto during her own invuln, she's given a chunk of damage to compensate for that down-time in DPS, but also giving opponents a way to avoid the damage from her ult. When she uses it on an ally, she's sacrificing her own survivability to help them survive, or utilize it in an offensive capability as more than just a panic ult.
By adding in the burst at the end, she's given counterplay to her invulnerability in 1v1s, and given new tools in how to successfully help a teamfight in more ways than just "I make them not die", at the exchange of using on of her main defense tools offensively, leaving herself vulnerable.
But addressing your post as a whole, as in your message you're trying to get across. We have very few true DPS champions, whose contribution is literally only DPS. Some of them, like Kallista and Xayah, are held behind some sort of gate/counterplay, Kallista needing 40 spears in you, Xayah needing her feathers. They present a challenge you need to circumvent, and reward you with achieving it, providing a high moment for the player who accomplishes it. The counterplay behind these moments is the real important thing, because let me explain your last point in practice.
If you're a DPS, and we remove these high moment type bursts, then at what point does simple steroids that enable DPS (AS or AD) result in the very same problem you're trying to avoid. Pump too much power into their steroids, and at what point does that DPS become burst regardless? Too strong an AD steroid, you're now doing burst per auto (Hello Draven). Too much AS, you're now plowing ahead with On-hits or insane other combinations where you're attacking fast enough for it to be burst (Hello Pre-reverted rework Kog'maw).
Honestly, we do need those burst, high moments for DPS champions. If you focus people too narrowly into DPS, issues arrive elsewhere, so having these rewardable tests in champions kits no only add in skill floors and ceilings to DPS auto champions is for the most part healthier because of the counterplay involved. As a concept, it is perfectly fine for this to be a thing. The only real issue is the same issue people keep complaining about on the boards in general: Too much damage in general.
Last two things I wanna throw in at the end: We've always had very few pure DPS champions in the game, they've always had chunks of damage in their kits. Even our most acclaimed DPS champions like Ashe or Twitch, have strong points of burst in their kit, be it her 5 stacked Volley, or Twitch's expunge or ult. Secondly, if you want to be a DPS, you need to kill tanks. If you need to kill tanks, you need the damage to do so, and the AS to do so. At somepoint, this overlaps with killing squishy champs too fast, because the damage needed to kill tanks is hitting squishies who don't have that mitigation.
If you focus too much solely on DPS, then you create more of those situations, because if they're fully designed with their power budget packed into DPS, they will bleed on over into being bursty against squishies and not tanks. By moving that power into more counterable moments of higher power, that can be dodged/counterplayed, then you don't have the DPS with autos alone to kill squishies fast, but successfully playing your champion will allow you to kill anyone faster and first.
: "Welcome to the club!"
Magic damage critical strikes
this item and it's build items are too burst-y to start with. 
