While it may seem obvious now that the phrase 'tan your hide' "takes its origins from the way that leather is processed", the historical evidence suggests that, insofar as the phrase does take its origins from that process, that is the least significant contribution to the original meaning. The context of early uses suggests, rather, that the phrase is an elided version of this claim:
I'm gonna flay you, then tan the hide.
Early uses do not necessarily express the first part ("flay you") outright; it was (and is), however, understood that a 'hide' cannot be tanned until the animal has been skinned.
The context of the first attestation in the OED,
Let not your Worships thick skin be too sensible that we thus Tan your Hide.
[from around 1670, in Expostulatory letter, A mild, from the plaindealing farmers of the neighbouring villages to the men of Buckingham]
is not available to me, so I must set it aside without comment. The second OED attestation, from the 1732 play The devil to pay (Charles Coffey),
...to Work, to Work, come and spin, you Drab, or I'll tan your Hide for you....
follows in the train of an earlier beating:
Lady. ...What Rascal are you?
Job. This is amazing, I never heard such Words from her before. If I take my Strap to you, I'll make you know your Husband....
Job. ...to Work, to Work, come and spin, you Drab, or I'll tan your Hide for you....
Coffey's play The devil to pay itself follows, too closely, in the train of an earlier beating, as encountered in The devil of a wife, a play by Thomas Jevon. Jevon died in 1688; his play was published in 1735, yet Coffey's 1732 play The devil to pay obviously owes...overmuch...to Jevon's play:
Job. Come, come, you Quean, I'll make you leave your fooling, come to your Spindle, or else I'll lam your Hide, you were ne'er lamm'd so since you were an Inch long.
In Jevon's play, "lam" (and "lamm'd") are used with the meaning shown in the 1825 A glossary of north country words in use, by John Trotter Brockett:
Lam, Lamb, to beat soundly. "Aw'l lamb yor hide."
"Lamb them, lads; lamb them!" — a cant phrase of the time, derived from the fate of Dr. Lambe, an astrologer and quack, who was knocked on the head by the rabble in Charles the First's time.
Peveril of the Peak, vol. iv. p. 132.
Plainly, 'tan' in the early use found in Coffey's play is being employed with the metonymic sense of 'beat'; already, before 1732, little of the literal sense of 'to tan', that is, "to cure leather with tannins", remains in Coffey's use, which seems to have been, um, 'lifted', more or less directly, from Jevon's earlier (before 1688) use 'lam' of in the sense of "beat".
The same metonymic distillation of 'tan' into the sense of 'beat' as was seen in Coffey's 1732 play can be found in another 1732 publication, a translation of Molière's The Miser, where the translation of je vous rosserai is "I shall tan your Hide." The French verb rosserai translates literally as "whip", "flog", "trounce", "beat up", etc. It is, however, beyond my expertise to say which meaning was likely in French of the 1600s.
Doubling back to Brockett's 1825 glossary, and an 1830 publication by Robert Forby (Vocabulary of East Anglia, a vocabulary which the title page advertises as having been collected in the last two decades of the 1700s), I observe that two other survivals (along with 'tan your hide' and 'lam') from the 18th century suggest the close association of 'tan' with 'flay', 'flog', 'thresh' — that is, the 'tan' in 'tan your hide' is the promised result after the acquisition of your hide by a severe beating. Those survivals are 'to hide' and 'a hiding', in the sense of 'to beat' and 'a beating' respectively.
Hide, to beat. "I'll hide your jacket."
[From Brockett, 1825: note that 'jacket' in Brockett's quote is used in the sense of "skin".]
HIDE, v. to thresh; to curry the hide. BR.
HIDING, s. a beating.
[From Forby, 1830: Forby's "BR." credits Brockett's definition; 'thresh' is what OED says is the "earlier and etymological form of THRASH v."]
None of the foregoing is conclusive with respect to my claim that, after examining the historical evidence given above and more, 'tan your hide' referred not so much to discoloration of skin as to a beating so severe as to flay the recipient, followed by a promise to tan the hide removed by the flaying. Either explanation of the origin of the term (that it refers to skinning followed by curing, or that it refers to simple discoloring) might serve as well as the other; however, in the absence of direct historical evidence contradicting my observations, mine is the more likely explanation of early use of the phrase.