Following "Arbitrage Theory in Continuous Time" by Thomas Bjork, a standard one-factor short-rate model is of the form \begin{align*} dr_t = \mu(t,r_t)dt + \sigma(t,r_t)dW_t. \end{align*} The only exogeneously given asset is then the locally risk-free money account with dynamics \begin{align*} dB_t = r_tB_tdt, \quad\text{ or }\quad B_t = e^{\int_0^tr_sds}, \end{align*} which can be shown to be equivalent to investing in a self-financing rolling over trading strategy, which at each time $t$ consists entirely of bonds with maturity at $t+dt$.
Now, from the $r_t$ dynamics one can derive the dynamics for the price of a $T$-bond, $p(t,T)$: \begin{align*} \frac{dp(t,T)}{p(t,T)} = \alpha_T(t)tdt + \sigma_T(t)dW_t, \end{align*} and the punchline is that for any maturities $S>t$ and $T>t$: \begin{align*} \frac{\alpha_T(t)-r_t}{\sigma_T(t)}=\frac{\alpha_S(t)-r_t}{\sigma_S(t)}, \end{align*} which means that the local risk-premium for exposing yourself to interest-rate risk is the same whether you invest in a bond with maturity $S$ or maturity $T$.
I understand the mathematics, but I am looking for an intuitive answer to the following question:
Why is there a risk-premium for investing in $T$-bonds?
Why don't we have $\alpha_T(t)=r_t$? Intuitively I feel like the $T$-bonds are just as locally risk-free as the money account, which, as mentioned above, is just like investing continuously in bonds that are about to mature.