  We extract the top-quark mass value in the on-shell renormalization scheme
from the comparison of theoretical predictions for $pp \rightarrow t\bar{t} +
X$ at next-to-next-to-leading order (NNLO) QCD accuracy with experimental data
collected by the ATLAS and CMS collaborations for absolute total, normalized
single-differential and double-differential cross-sections during Run 1, Run 2
and the ongoing Run 3 at the Large Hadron Collider (LHC). For the theory
computations of heavy-quark pair-production we use the MATRIX framework,
interfaced to PineAPPL for the generation of grids of theory predictions, which
can be efficiently used a-posteriori during the fit, performed within xFitter.
We take several state-of-the-art parton distribution functions (PDFs) as input
for the fit and evaluate their associated uncertainties, as well as the
uncertainties arising from renormalization and factorization scale variation.
Fit uncertainties related to the datasets are also part of the extracted
uncertainty of the top-quark mass and turn out to be of similar size as the
combined scale and PDF uncertainty. Fit results from different PDF sets agree
among each other within 1$\sigma$ uncertainty, whereas some datasets related to
$t\bar{t}$ decay in different channels (dileptonic vs. semileptonic) point
towards top-quark mass values in slight tension among each other, although
still compatible within $2.5 \sigma$ accuracy. Our results are compatible with
the PDG 2022 top-quark pole-mass value. Our work opens the road towards more
complex simultaneous NNLO fits of PDFs, the strong coupling $\alpha_s(M_Z)$ and
the top-quark mass, using the currently most precise experimental data on
$t\bar{t} + X$ total and multi-differential cross-sections from the LHC.
