Prior to burst windows all being synchronized, I simply cleared this sort of content without caring about burst windows (aside from potions if the party was using/coordinating them) but I'm bias here as a tank with no raid-wide damage buffs.

I don't know that the fights were designed to be synchronized particularly in the past, because players that synchronized and optimized everything found DPS checks overly trivial compared to regular players (at least from what I used to see hardcore statics say). But now that they are synchronized by design, they can scale the content accordingly (but the first raid tier, which we're currently on, is usually more lax in all respects).

That said, you'd always want some of the following to actually clear successfully:
  • Players that can actually do their rotation well.
  • Food.
  • Tinctures.
  • Materia melds.
  • Gear higher than the content.
Again, I said the word "some". You don't necessarily need all of these things but the more that you have the more likely the party would clear regardless of things like burst windows. And obviously the DPS checks get rougher on the final floors.

In EW, all the burst windows were synchronized by design, thus allowing them to scale the fights with synchronized bursts in mind. So even if you're not trying to synchronize your burst, it will be anyway now in most cases, making the question a bit pointless in my opinion.

In my experience, the first Savage floor typically dies about half way through the fight by the end of the tier due to everyone being ilvl capped. It's crazy how much of it usually gets skipped. The gear increases have a huge impact on their own, so it's easy for players to jump in a bit later in the tier for an easier time with DPS.

Regarding your points:
  • Jobs are homogenous because when they are not, people complain "my job doesn't have X ability that other job has". Gap closers are an easy example but there are plenty of others.
  • Synchronized burst windows do cause some of the homogenization and this was made worse in EW with the alignment of burst windows to 1 or 2 minute intervals. The easy solution, in my opinion, would be removing raid-wide damage buffs altogether and making damage buffs not multiplicative, so that job design isn't constrained as much.
  • The DPS checks aren't really that hard usually, unless the party sucks or has composition issues (ie. all the lowest DPS jobs in one party together with lack of raid buffs). I've often killed savage floors with milliseconds to enrage early in the tier, and sometimes there are wipes the few pulls before it, but guess what? You just all think "what could I do better to get a bit more damage in?" and if all 8 of you do that, then you beat it. The DPS checks get way easier as weeks go by due to tomestone gear/normal raid weapon etc.
  • PF prog will always be unsatisfactory unless they literally make it as hard as Ifrit normal, because it's PF.
The burst windows being synchronized in EW was obviously meant to make casual players do what hardcore statics were doing already, without realizing it, so that the DPS checks weren't easier for those synchronizing them. The side effect was us reaching a point where world racers actually couldn't clear the DPS check in P8S - which might not have happened pre-EW even if they over-estimated HP due to them not accounting for raid buffs being synchronized as much.

I don't really get all your talk about third-party tools. You don't need them to clear content nor to improve, provided you understand basic things about the game like snapshotting and uptime. EU players tend to demand them a lot more than NA in my experience, so I can see why it's bothering you, but it's not as much of an issue in the NA region.