Add a new skill line to each Class as you rework them.
Pure Class players will be allowed to run four Class skill lines.
Non-Pure Class players will be limited to running three Class skill lines.
Shift Class Sets to require four pieces instead of five to get the five piece bonus if the player is Pure Class.
Obviously, this idea is DOA if the memory supply is still heavily limited as you would be needing to hold a bunch of new abilities in memory. This approach will also obviously take more work to implement and will put a bit of a damper on the amount of abilities for future classes as some of the abilities you might add to existing classes might have instead gone on new classes in the future.
The upside to the approach is that it would enable reworks to more easily feel like they are adding more than they take away as you are literally adding abilities and it would allow for the design approach to take a somewhat lighter touch as you aren't competing against lines in as much of a one to one fashion.
For example, the next two reworks coming are the Warden and the Sorcerer.
The Warden does Magic Damage, Frost Damage, Poison Damage, and Bleed Damage. It uses a bear, creatures from Vvardenfell, mushrooms, vines, flowers, and ice.
I think it would be easier to deal with reworking Warden if you could just go Winter + Spring + Summer + Autumn with some added skill styles for people that wanted to lean a bit harder into a specific theme rather than trying to force all of that to fit a clean theme in the existing three lines. You'd have enough skills that players wouldn't be stuck leaning too deeply into elements they didn't care about.
The Sorcerer has a more limited array of damage types than the Warden but it does have a pet problem. Some of the players that play Sorcerer love pets and some of the players that play Sorcerer hate pets. This leaves you with a bit of a problem as one of your three class skill lines for Sorcerer has multiple pet abilities meaning that it isn't going to be a good for DPS that don't like pets. The easy way to dance around the issue is to split the pet abilities across multiple lines and to raise the number of DPS abilities the Sorcerer has that are effective alternatives which would be easier to do if the class had a fourth line (Thaumaturgy has had some relatively varied definitions so you could likely justify moving a wide variety of Sorcerer abilities into it).
Another issue that you hit with the current three class skill line approach is that to make the lines competitive with subclassing you need to make multiple abilities in a line stronger than non-class abilities to avoid players simply selecting lines based off of passives or a single strong ability. This can put players into a position where the effective build options end up being a bit limited as non-class abilities fall behind.
For example, Vampirism has an ability that increases your weapon/spell damage at the cost of constantly draining your health and has an ability that restores a % of your missing health and a portion of your missing stamina that does a little bit of single target damage. The new DK has an ability that increases your damage done just for using DK Class abilities and that ability also functions as an attack rather than just as a buff like the Vampire ability does. The new DK has an ability that restores a % of your missing Magicka, Stamina, and Health that does a decent amount of AOE damage and the ability doesn’t require channeling for multiple seconds unlike the Vampire one.
Class abilities also tend to lean towards primarily being Magicka slanted when it comes to abilities and even when the cost isn’t Magicka slanted the abilities style frequently is. Yeah, a DK or Arcanist might be consuming more Stamina than Magicka but, visually the abilities tend to align more with Magic than they do with anything mundane. Some people like the more mundane parts of the Elder Scrolls as much or more than the Magical parts and shifting too much towards a flashy magical focus can potentially put a damper on the overall appeal.
Finally, you also hit the question of how much you can really make the classes feel unique in the long run if you are trying to use five regular abilities and one ultimate plus morphs in 21 lines to all provide value in a comparable fashion to Tanking, Healing, Damage Dealing, and PvP Roles. If we look at the DK rework, we can already see some redundancy with existing abilities. The new DK Morph of Cauterize is basically just a better version of the Sorcerer Morph Power Surge with a different buff. The new DK Standard of Might runs kinda close to the Arcanist ability Glyphic of the Tides.