Why is "desperacy" not an English word?

Solution 1:

I wonder where the OP has heard desperacy much more than desperation.

The OED says:

desperacy, n. Obs.

Etymology: < desperate adj., n., and adv.: see ‑acy suffix.

Desperateness, desperation.

• 1629 J. Gaule Practique Theories Christs Predict. 11
  Downe to the nethermost depth beyond recouerie: Let vs there take our portion of desperacie.
• 1798 Ann. Reg. (Otridge ed.) Hist. Europe 155/2
  Such deeds of desperacy and revenge.
• 1800 W. E. J. Obi 231
  Deeds of desperacy and cruelty.

which suggest that it was a used word, about two centuries ago. Ngram can find no example, nor can Google Books, nor can COCA.

It is not that that desperacy is not an English word. It is not a current English word.

Solution 2:

Desperacy is “not a word”¹ because just plain despair alone plenty suffices as a noun, without any redundantly tedious nominalizationalizing suffixes and postfixes and add-on-the-enders.

The take-away lesson here is that sometimes, perhaps even often, it is far better to trim things off instead of tack them on.

1. _{That is, not a word attested² by the Oxford English Dictionary.}

2. _{That is, not a word attested as being current in contemporary English.
   However, the OED does attest an obsolete word †ˈdesperacy,
   and for which it provides no citation later than from the 18^th century.}

As for the second part of your question, here are some attested triples that may interest you:

• adequacy, adequate, adequation
• advocacy, advocate, advocation
• alternacy, alternate, alternation
• articulacy, articulate, articulation
• automacy, automate, automation
• complicacy, complicate, complication
• confederacy, confederate, confederation
• conjugacy, conjugate, conjugation
• degeneracy, degenerate, degeneration
• delegacy, delegate, delegation
• determinacy, determinate, determination
• effeminacy, effeminate, effemination
• episcopacy, episcopate, episcopation
• illegitimacy, illegitimate, illegitimation
• inadequacy, inadequate, inadequation
• inarticulacy, inarticulate, inarticulation
• inchoacy, inchoate, inchoation
• indeterminacy, indeterminate, indetermination
• inebriacy, inebriate, inebriation
• inordinacy, inordinate, inordination
• intermediacy, intermediate, intermediation
• intestacy, intestate, intestation
• intimacy, intimate, intimation
• itineracy, itinerate, itineration
• legitimacy, legitimate, legitimation
• lunacy, lunate, lunation
• numeracy, numerate, numeration
• obduracy, obdurate, obduration
• oracy, orate, oration
• prelacy, prelate, prelation
• privacy, private, privation
• profligacy, profligate, profligation
• prolificacy, prolificate, prolification
• regeneracy, regenerate, regeneration
• reprobacy, reprobate, reprobation
• subordinacy, subordinate, subordination
• surrogacy, surrogate, surrogation
• testacy, testate, testation
• ultimacy, ultimate, ultimation
• unregeneracy, unregenerate, unregeneration