Generally I don’t think grid-filling for the sake of grid-filling leads to the best designs. But it’s still an interesting exercise. I agree with cleric -> paladin -> fighter and druid -> ranger -> barbarian, but wizard -> artificer -> rogue feels weird to me. I could see wizard -> bard -> rogue working if bards were half-casters in 5e, but since they’re full-casters, I think chart might look something like…
Full Caster Half-Caster Non-Caster Bard Arcane Trickster Rogue Cleric Paladin Fighter Druid Ranger Barbarian Sorcerer Mystic Monk Warlock Hexblade Assassin Wizard Artificer Tinker
Arcane trickster is of course a rogue subclass in 5e, but if we’re committing to this grid idea I would expand the concept into a full class with half casting (and maybe half sneak attack progression?)
Similarly, hexblade is a warlock subclass, but it has always been a weird one, and I think would be better served as a “half pact magic caster.” For the non-caster in this lineup I went with assassin despite that being another rogue subclass already. But in trying to come up with a non-magical class with a thematic link to warlocks and hexblades, I imagined a character bound in service by a non-magical “pact” to a temporal “patron”, and I realized a contract killer fit that description perfectly.
I’m using “mystic” as the name for the half-caster between Sorcerer and Monk, but I imagine it being quite different than the mystic from Unearthed Arcana. Rather than the equivalent of a full-caster in a full psionic casting-but-not-casting subsystem, I imagine this mystic as bridging the gap between sorcerer and monk through the common theme of inner power. Where the sorcerer’s innate power is purely metaphysical and the monk channels their inner power through their physical body, the mystic would use their mind as a conduit between the two. Sorcerers and monks also both use a power point system, so I imagine the Mystic would do so as well.
The tinker is just a straightforward non-magical artificer. They make cool gadgets and they don’t cast spells. Not very inspiring, but there you have it.
I like the tinker idea. Engineer or mechanic might fit in there too. Reminds me of an old RPG video game with a smith class.
I had the rogue in with the wizard and artificer because the rogue is the go-to class for versatility in the martial classes, often relies on wits or intellect (at least in stories), and often uses tools for effect (think flash bombs grenades, caltrops, ball bearings).
The concern with the bard to trickster to rogue is the seems to point to a specific style of bard rather than being more representative. Not that it doesn't work like that because it is a common thematic bard but it's not my personal typical bard.

It's a nice list.
Warlock is only a full caster in the sense that they get spells of every level; in many other respects they are half-casters, so I put n/a in the half-caster spot for that line (the table editor doesn't do spanning cells).
Warlocks gain spell levels at the same rate as other spell casters as well and using the short rests can cast more spells per day. They are a variation on a full caster by splitting resources. The ability to spam spells is metered on them and that's the only real difference.
Having said that, if I were to do a half-caster on the same model I would probably limit it to 1 slot per short rest until 11th level and then increase it to 2 slots, not give cantrips, add a fighting style, customize a list of invocations with half the progression warlocks have, and then add a few thematic subclass abilities. Possible a couple of class abilities, but definitely no arcanum.
I consider warlord a subtype of fighter that the 5e chassis will never be capable of satisfying the 4e fans of the class. The concept is sound but there's only ever been that design structure and it doesn't fall back in to 5e terms as well as updating older classes.
Oh, my god, I forgot about them! That could totally replace “assassin” as the non-caster on the warlock row in my grid!
I forgot about them at first too. They do fit.