Any reason private keys are not just numbers between 0 and 2²⁵⁶-1 ?

The range of valid private keys is governed by the secp256k1 ECDSA standard used by Bitcoin.

As excerpted from Standards:

The elliptic curve domain parameters over Fp associated with a Koblitz curve secp256k1 are specified by the sextuple T = (p,a,b,G,n,h) where the finite field Fp is defined by:

    p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F
    = 2256 - 232 - 29 - 28 - 27 - 26 - 24 - 1

The curve E: y2 = x3+ax+b over Fp is defined by:

    a = 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000
    b = 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000007

The base point G in compressed form is:

    G = 02 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798

and in uncompressed form is:

    G = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8

Finally the order n of G and the cofactor are:

    n = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141
    h = 01