That sort of over-simplification only hurts the ability to actually look at and understand the Champion. It's like when Galio was reworked, and people only looked at Hero's Entrance and spent the entirety of the PBE cycle going "It's just a better/shittier version of Shen".
Besides, you're ignoring the larger mechanical differences between each of the abilities you listed (some of which, by the way, aren't even remotely accurate)
Tahm Kench's passive isn't his Gray Health. That would be his E, Thick Skin. His Passive is An Acquired Taste, which allows him to stack up Acquired Taste on a Champion to augment/allow his abilities.
Beyond that, though, while Gift of the Drowned Ones is ostensibly just the passive half of Thick Skin (Gray Health conversion), it requires that Pyke not be seen by enemies to allow the healing conversion to happen.
Bone Skewer (Harpoon) is less akin to Blitzcrank's Rocket Grab and more like a mix of Varus' Piercing Arrow and Taliyah's Seismic Shove. It requires charging, and it will only throw a target a set distance in Pyke's direction - at max range, it would draw them into, say, basic attack range, but at point-blank, it will work more like Singed's Fling, tossing an enemy behind him.
(which, is why I used Seismic Shove as the primary reference for it, as it's a skillshot and the distance thrown is based on positioning.)
Bone Skewer (Dagger)...I don't see how you relate it to Swain's Nevermove. Nevermove doesn't slow, and it roots on the way back. Bone Skewer is more akin to a mix of Yasuo's Steel Tempest (non-empowered) and Darius' Cripple. It's a line skillshot thrust that deals damage and applies a heavy slow.
Ghostwater Dive is less like Evelynn's Demon Shade and more like Twitch's Ambush, sans attack speed steroid and reset, and with diminishing movespeed.
Phantom Undertow is more like Shen's Shadow Dash + Twilight Assault combo, except the image moving triggers the stun and does so on its own. Even still, that's a pretty bad attempt at describing it, but it's certainly more accurate than using Irelia's Flawless Duet.
Death From Below is about the closest to accurate - it is somewhat like Darius' Noxian Guillotine, but also not like it at the same time. It's an AoE Skillshot that you can actually dodge, and the execute is less based on stacks (like Darius' Hemorrhage) and more reliant on an execute threshold (like Evelynn's Last Caress).
Even still, likening his abilities to other Champion abilities only detracts from the ability to understand Pyke's kit, especially when those example Champions are all over the place in role (Varus as a bottom laner, Taliyah and Swain as mid laners, Shen and Darius as Top laners, Tahm Kench and Blitzcrank as Supports...Etc.)
When we look at new Champions (or VGU's), we should only look at Champions abilities as being similar to others as a point of reference for understanding how to use them, not as a point of reference to infer their role.
Pyke is a Support-Assassin. He has some of the advantages and disadvantages of Assassins (high mobility, stealth, burst damage, but very lightweight and easy to kill) with those of Supports (Crowd control, ability to complement or uplift a given teammate, utility, but lack of extra ability without a teammate).