This post is going to be mostly from a PvE perspective. I'm going to start by clarifying that I don't hate light attack weaving. As a playstyle, I think it's got a decent flow and thematically it fits nicely with a sustained, constant-pressure DPS. That said, I feel there are a couple major issues with the current combat system that could be helped by further expanding it beyond this one mechanical playstyle.

Why LA Weaving is so good (If you didn't already know): LA Weaving became popular for one reason. Simply, it is the fact that basic attacks are on a separate yet similar cooldown from skills, so it's ALWAYS better to perform your actions as LA>Skill, to the tempo of the GCD.

Issues with LA Weaving: Perhaps my biggest issue with LA weaving is how it's changed our view on light attacks. When you look at most action combat systems, basic light attacks are your core, staple attacks for non-caster builds. They are your main spammable, equivalent to your primary fire in an FPS game, often featuring at least a basic combo system to fluidly chain multiple light attacks together. In ESO however, with LA weaving, light attacks have become closer to the MOBA/Tab-targeted-style of Auto Attacks, treated as just another passive damage source to keep up with the rest of your rotation. It is even to the point where ZoS started creating spammable skills, where we give up a skill slot for something that basic attacks should already provide. Additionally, as LA weaving was originally a bug (No, it was not a deliberate design choice), the animation cancelling involved allows you to essentially bypass the actual attack altogether, and simply get the extra damage with no cost (While also looking horrendous to new players). This greatly cheapens the attack itself, taking it from an active, deliberate action, to one you mostly forget once you've learned how to use it best. This seems particularly unsuitable for a game with a limited hotbar, where every action you have access to should feel impactful.
My other major issue with LA weaving is how overbearing it has become. While there are some key changes that really hurt class and build diversity, I feel that LA Weaving has had one of the greatest impacts. Skills and sets were changed to fit this style, and basic attack cooldowns across weapon types were flattened so they all worked with weaving. It is very difficult to create a unique character when every single element of the game is catered to one specific way of playing.

Ways to add new mechanics: There are a few things found throughout other action combat games (As well as other genres) that I think would really help break up the monotony of the LA weaving meta. Of course, as I am simply a player and not a developer for ZOS, I have no idea how difficult any of these would be to implement.

- Combos: This, to me, is the most obvious and likely most effective addition. From Dark Souls, God of War, or even a game like Hades, to other MMOs like Tera, Black Desert or Blade and Soul, combo systems have been proven to work incredibly well with action combat. The technical requirements already exist in the game; skills like Crystal Fragments and Molten Whip show the game can already track the last used skill and adjust other skills based on its characteristics. There is also the Molag Kena set, which procs after using two consecutive light attacks, indicating a chaining functionality exists as well. Such a system could be used in a myriad of ways, such as buffing consecutive light attacks, creating "finisher" attacks by incorporating heavy attacks in the chain, or creating skill combos that are disrupted by using basic attacks between skills. There are so many options here, and any one of them could be balanced to provide a competitive alternative to LA weaving.

- Cast Times: I know a lot of people really despise cast times, but used effectively I think they could provide another avenue for damage dealers who prefer a more tactical approach to combat. More traditional MMOs often have classes (usually mages) where the player spends a fair deal of time either immobilized or slowed down by casting, and they're usually designed to reward high-risk, high-reward playstyles that require the player to make tactical decisions about the best time to cast a skill based on their positioning and the boss's mechanics. If you're already typing your angry rant about how terrible cast times are, the important thing to remember here is that we're trying to ADD options for players who want them, without removing options for players who still prefer a higher-APM playstyle. Side Note: Cast times also provide a really effective way to balance high-impact skills, such as heavy CC or damage skills, the former of which is a bit rampant and the latter of which doesn't really exist anymore.

- Restore unique weapon cooldowns: I personally was really disappointed when they changed all basic attacks to use the same cooldown. Originally, you could tell that Two-Handed was designed to be the slower, more powerful melee option, while Dual-Wield offered speed and flexibility. Now the two feel identical, even down to each tree using a cast-time/channeled ability as their main spammable. By having a gap between weapon types, you can create options like using daggers to get two LAs in between each GCD, or using a warhammer to pummel through multiple enemies in your face.

- Restore LA/Cast speed modifiers: One of the most unique mechanics in the game was the old Nightblade skill Haste that increased your LA speed by a significant amount. Unfortunately, the skill was replaced with one that arguably would have been its best compliment, Grim Focus. The two together would really cement that bursty, assassination-style of damage that the skill line is named for, with Haste allowing you to rapidly generate focus stacks for the big burst. In the context of this thread, such effects would give players a lot of flexibility in creating a unique pace for their builds, rather than the strict one-second metronome we have now (Literally, it's even an add-on).

I'm sure there are several other routes the devs could take to address this, any of which I'd be happy to read about. I'd like to clarify once again that I don't think we need to remove LA weaving, but rather provide alternate playstyles that can even come close in terms of DPS. If you read this far, thanks for taking the time. If you have a comment/critique/suggestion, let me know.