Being model-independent, other than assuming $\Omega_k=0$ (zero curvature of the universe), supernovae measurement of $H_0$ does not need the input of $\Omega_m$, $\Omega_r$, or $\Omega_{\Lambda}$.
Supernovae methods directly measure $H_0$ $$ H_0 = \frac{\dot{a}(t)}{a(t)}|_{t=t_0} $$ via (see here)
- Red shift to infer the speed of the receding supernovae-hosting galaxies
- Cosmic ladders like Cepheids to infer distance
On the other hand, $H_0$ from Planck is model dependent, since it needs the input of $\Omega_m$, $\Omega_r$, and $\Omega_{\Lambda}$ at red shift $z\sim 1100$ to derive the Hubble parameter at red shift $z=0$ by integrating the Friedmann equation all the way from $z\sim 1100$ to $z=0$ as shown above in the OP.
Generally speaking, direct measurements are more trustworthy. Supernovae measurements are actually corroborated by other direct measurement methods (see here, here, here, and here). Barring systemic bias, the culprit and the root cause of "Hubble tension" is most likely the standard cosmology model $\Lambda$CDM (see Is standard cosmology $\Lambda$CDM currently in deep trouble?). Either
- $\Omega_m$, $\Omega_r$, or $\Omega_{\Lambda}$ is possibly off, they are not what we think they are. To quote Donald (another Donald) "there are known knowns...there are known unknowns...there are also unknown unknowns". One might consider adding more "dark" elements into the mixture like the Kafkaesque scenario of "dark interaction between dark matter and dark energy" (I am not making this up. See here).
- or there is something wrong with Friedmann equation and for that matter Einstein's gravity equations.
The Jury is still out. And it's an opportune time to be a cosmologist now, since historically "cloud over physics" is a harbinger of phenomenal science breakthroughs.
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Added note:
There is an interesting Youtube video on the "The crisis in cosmology" (see here).