What exactly is a 'chiral centre' in a carbon compound? The typical progression (in books, YouTube videos) of explanation of chiral carbons in a compound is as follows:

Chiral compounds are non-superimposable by their mirror-image;
It is easily seen that only carbon atoms with all substituents different are chiral;
In a carbon compound with more than one atom, (2.) is exactly how you check for chirality;
It isn't necessary for a compound to be chiral despite having chiral carbons.

It seems to me that something between (2) and (3) is being eaten up. How does a chiral carbon in an chiral compound show the non-superimposability property? How does it satisfy the basic definition of chirality?
Skipping axial and helical chirality altogether, no to point #2: there are examples of atom centred chirality without a carbon atom in the centre.  The careful oxidation of thioethers may yield chiral sulfoxides:

(Han et al., Chem. Soc. Rev. 2018, 47, 1307-1350, doi 10.1039/C6CS00703A).
Here, chiral requires just the two carbon substitutents to differ, because the third is oxygen, and the fourth the lone electron pair.  Thus, applying CIP rules as usual where the lone electron pair has the lowest priority, DMSO is not chiral, contrary to methyl phenylsulfoxide which is.
Just as with stereogenic centres around carbon, deprotonations on the adjacent prochiral carbon atom may proceed with preference for one hydrogen over the other:

(Solladié, Synthesis 1981, 185-196, doi 10.1055/s-1981-29378).