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We investigate two new observational perspectives in the context of torsional gravitational modification of general relativity, i.e., the $f(T)$ gravity: i) We use Pantheon data of type Ia supernovae motivated by a time variation of the Newton's constant on the supernovae distance modulus relation, and find that a joint analysis with Baryon Acoustic Oscillations (BAO) and Big Bang Nucleosynthesis (BBN), i.e., Pantheon+BAO+BBN, provides constraints on the effective free parameter of the theory to be well compatible with the $Λ$CDM prediction; ii) We present the framework of $f(T)$ gravity at the level of linear perturbations with the phenomenological functions, namely the effective gravitational coupling $μ$ and the light deflection parameter $Σ$, which are commonly used to parameterize possible modifications of the Poisson equation relating the matter density contrast to the lensing and the Newtonian potentials, respectively. We use the available Cosmic Microwave Background (CMB) data sets from the Planck 2018 release to constrain the free parameters of the $f(T)$ power-law gravity and $Λ$CDM models. We find that CMB data, and its joint analyses with Pantheon and BAO data constrain the $f(T)$ power-law gravity scenario to be practically indistinguishable from the $Λ$CDM model. We obtain the strongest limits ever reported on $f(T)$ power-law gravity scenario at the cosmological level.