We trivially have that:
$$\frac{Z(t_0,t_1)}{Z(t_0,t_2)}=1+\tau L(t_0,t_1,t_2)$$
Where $L(t_0,t_1,t_2)$ is the forward Libor between $t_1$ and $t_2$, as of $t_0$.
Simply inverting this relationship then yields:
$$\frac{Z(t_0,t_2)}{Z(t_0,t_1)}=\frac{1}{1+\tau L(t_0,t_1,t_2)}$$
Could one interpret $\frac{1}{1+\tau L(t_0,t_1,t_2)}$ as a forward starting zero-coupon bond between $t_1$ and $t_2$, as of $t_0$?
I.e.:
$$\frac{Z(t_0,t_2)}{Z(t_0,t_1)}=Z(t_0,t_1,t_2)$$
If the above is true, then suppose we want to value a Caplet "set in arrears" (i.e. pay-off described in my last question).
This caplet pays $(L(t_1,t_1,t_2)-K)^{+}$ at time $t_1$. Valuing this caplet at $t_0$, choosing $Z(t_0,t_2)$ as Numeraire, we have:
$$C(t_0, T=t_1)=Z(t_0,t_2)\mathbb{E}^{t_2}_{t_0}\left[\frac{(L(t_1,t_1,t_2)-K)^{+}}{Z(t_1,t_2)}\right]$$
Using the identity:
$$Z(t_0,t_2)=Z(t_0,t_1)Z(t_0,t_1,t_2)$$
I get:
$$C(t_0, T=t_1)=Z(t_0,t_1)Z(t_0,t_1,t_2)\mathbb{E}^{t_2}_{t_0}\left[\frac{(L(t_1,t_1,t_2)-K)^{+}}{Z(t_1,t_2)}\right]=\\=Z(t_0,t_1)\mathbb{E}^{t_2}_{t_0}\left[(L(t_1,t_1,t_2)-K)^{+}\right]$$
And the problem at hand now seems trivial, since $L(t_1,t_1,t_2)$ is a martingale under $Z(t_0,t_2)$.
The above cannot be correct, since the answer is different to what @Gordon derived in my previous question linked above. So where have I gone wrong here?