iLv needs to change

[Felessan]

Item Level should be the first stat people can look at to judge their relative strength, but as is, it's just a prerequisite stat people look at to allow them to enter instances or join Party Finder groups, or compare as an epeen between people with optimal weapons.

ILVL requirement only ensure that you are not too undergeared for this particular content, so making weapon requirements is pointless as even if it have greater weight in your performance it doesn't ensure in any way your success.

But then a problem still stands about what to do about this suddenly-common occurrence of big-armor, little-weapon players?

The biggest impact on performance is lack of skill, not a bad weapon.
I (relatively) often see how people with ILVL90-100 weapon outpefroms people with 120+ weapons just because they know their class and proper rotations.
A quick calculation gives me that at ILVL130 changing weapon from ILVL90 to ILVL135 improves damage by about 15% based on NIN stat weights. It's not that much - you can easily see in WoD for example how people have half of dps of the top spot with similar equipment just because they obviously either not trying or doing something wrong.

They can technically enter T13. But if they're a DPS, for example, their damage output will be about half or less of what someone with an i135 weapon and full i120 gear + one i130 ring (also amounting to an overall iLv of 123) can do.

This is not even remotely true. Just checked myself. With i55 weapon I have 72% of dps of i135 weapon (40% drop).
You really underestimating the combined power of other gear slots. Vast amount of primary and secondary stats far outweigh additional weapon damage stat weapons have.