For what it's worth, you're honestly not wrong about it being arbitrarily restrictive as far as I can consider it. It's a legacy system at the very least. I'd venture that it's just a holdover from early in the game's development where they more directly copied other successful MMOs and that it hasn't really been re-evaluated, unlike other parts of the same system. I don't think that they should continue to be done this way.
Hidden for a bunch of semi-relevant ramblings on how the system wound up this way and why it feels so weird in the current state especially:
Earlier MMOs tended to have much less to do, but they definitely wanted to retain monthly subscription numbers. The easiest way to design for that was to include tried-and-true time-sinks and time gating systems. Given raids were the primary way to get the best gear, it made sense economically to limit the amount of rewards players could get from them over time and via RNG, because for many players there wasn't else much to do once you got all the items you wanted.
Those raids probably were thought to be necessarily long to justify the weekly lockouts and monthly subscriptions, so they were filled with lots of walking and trash, and that reinforced the need for a weekly lockout from the player's view because progressing would be impossible if a raid's bosses all came back every few hours or a day or what have you. And, playing with huge groups of players is kind of the appeal of MMOs to begin with, so it makes sense that designers and players would expect the most challenging form of content to reflect that.
It was much the same when FFXIV launched. A tighter weekly cap of 300 tomestones and a weekly raid lockout were the things to keep players revisiting weekly if they didn't have any interest in leveling other classes or just socializing (no housing, Gold Saucer or glamour at this point, and there's maybe 4 total mounts in game). Reflective of WoW's raids, the raid duties were also designed to be progressed through like dungeons, having shortcuts that opened up as you progress (you can see this in many Bahamut's Coil and Alexander (normal) raids today still). They also required you and your party to meet in the overworld near a certain NPC to queue into the duties, not unlike how Deep Dungeons or Exploratory Zones still work. FFXIV's trash has mostly always been far more trivial than in other MMOs (usually to its credit), and most players don't like it at all as on top of that they don't even offer rewards, so player feedback eventually saw this downsized to now where a raid is indistinguishable from a trial once you're in (other than "difficulty," ostensibly).
Crucially, however, in their original system you could not go back to floors you had previously cleared for the week. I think this was their system of implementing the weekly loot lockout, because no other duty had something like this at the time (Labyrinth of the Ancients, which was introduced in 2.1, was the first duty to implement this), and I'm pretty sure you could not even re-enter floors you had completed for the week until Alexander Gordias (Savage) in patch 3.05.
When they introduced this, they also introduced the penalty for players that hadn't cleared to have someone in the party who had. The reason this was done was to reinforce ilvl progression on savage content, as they were much tighter at the time, normally requiring even the best groups that cleared within the first week to have players with Vitality-melded accessories to survive damage from the later raid floors. It also made it logistically harder for players to feed gear to specific players because you would need a team of alts to bring each person you were feeding gear to through each floor, and you couldn't predict what gear you'd end up with and just roll with team of whatever jobs got the most gear (this is an actual strategy that players have used and probably still use, though now savages aren't tuned so tightly so it'd be less common).
Additionally, drops were random up through 5.05, when raids started to drop coffers instead of job-specific gear, and 6.4 where they're now guaranteed to drop one of all possible coffers instead of potentially doubling-up on and missing certain slots each week.
Given loot is so much less random nowadays, I can't conceive of a logical reason that they would still enforce the 1-2-3-4 floor progression as the only time-gate on savage gear is now winning rolls against others (in PF groups) and the weekly lockout itself (floor 4 will still take the same amount of time for a static group to get everyone a mount as before at 8 weeks of clears).
The only thing I can think of is maybe they think it keeps raid/party finder more full of players with less floor skipping. But players have been sharing raid lockouts to skip unwanted floors since 2.0, or even "stealing" lockouts by joining a party finder and immediately leaving the duty, wasting the others' time.
The current system puts extra pressure on players whose statics are "done" gearing from earlier floors either to do this or, for their members that want gear for alternate jobs, to rush to clear floors with players they'd rather not play with or at times they don't prefer, and it also makes the players that are regularly clearing even more lopsided to the front of the week-- if players could go in any order, the playerbase available for folks still learning might be more evenly spread throughout the week for the above example alone, but also the rush wouldn't be as necessary as those same players could join whichever floor re-clear party in any order whenever it's convenient for them, not having to miss parties further along if they couldn't play earlier that week.
I'd love if they simply removed the weekly restriction on loot entirely given there's nothing players gain from that (that's obviously still all down to subscriptions), or at the very least the penalty for parties with members that have already cleared for the week.