No. There's nothing missing regarding the Planck units, they just reflect dimensional analysis. Quantum gravity isn't just dimensional analysis, just as electrodynamics isn't just dimensional analysis. Dimensional analysis implies that μ_0ε_0 has the dimensions of 1/velocity^2 and the numetical value of tgat velocity is that of light; but it's not possible from that information alone to deduce the Lorentz invariance and gauge invariance of electrodynamics.

Quantum gravity does not require the existence of particles of mass equal to the Planck mass.

Jamil Kooli , Stam Nicolis ,

as long as we can assume that the gravitational constant G actually IS constant, at least the Planck units for length, time and mass ARE a direct consequence:

We have the orbit velocity of a mass orbiting around M at distance r

v_orb² = G∙M/r

If we take v_orb to its absolute limit c, we get

c² = G∙M/r

Additionally, at quantum level, we have the equivalence of mass and its related oscillator's wavelength

m = h/(c∙λ) = ħ/(c∙r)

If we now put together the last two equations, we obtain

c² = G∙ħ/(c∙r²)

which is

r² = G∙ħ/c³,

which ist the smallest possible r.

This smallest r is the Planck length ℓₚₗ

ℓₚₗ = √(G∙ħ/c³)

Planck time is simply derived from

tₚₗ = ℓₚₗ/c.

Planck mass again using the oscillator's mass-wavelength equation above:

mₚₗ = ħ/(c∙ℓₚₗ) = √(ħc/G)

To:

“If quantum gravity is adopted, can we then speak of a possible rest mass of the Planck particle?”:

The rest mass, M, of the hypothetical [though rather probably really existent] “Planck particle” [more correctly “the Planck mass particle”] is determined by main Planck units: M=ћ/tP/c2, ћ is Planck constant, tP is the Planck time, [speed of light] c=lP/tP, lP is Planck length. I.e. while every particle has rest mass m=ћω.c2, where

[in the Shevchenko-Tokarevsky’s Planck scale informational physical model, in this case it is enough to read one of two main papers in https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics that all particles are some cyclic algorithms that tick with frequency ω; and so all particles have energies E=ћω and masses m=E/c2]

ω has maximally possible value =1/tP.

That’s all, in the rest relating to Gravity force, including to Gravity quantum nature, Planck mass particle doesn’t principally differ from any other particles, though in ths [quantum Gravity] case here is some essential point, more see SS post in https://www.researchgate.net/post/Do_you_believe_that_Planck_mass_and_Einstein_mcc_are_the_key_to_quantum_gravity

And to

“…If so, will we have the missing link of Planck units, in case there is indeed a lack of these units? ….”

- from the above it follows that for description of most of fundamentally universal basic points in Matter the Planck units ebove are sufficient, however for Matter be as it is it is necessary to have non-universal, but also fundamental, items - the [now known 4] fundamental Nature forces, actions of which are determind by the Forces strengths constants [more see in this case other main paper

https://www.researchgate.net/publication/383127718_The_Informational_Physical_Model_and_Fundamental_Problems_in_Physics “Mediation of the fundamental forces in complex systems”

, and this system of two types constants above looks as is completely sufficient.

Cheers