Tokyo update
Suppose we adopt the axiom that $\mathbb E(S_1)=(1+\epsilon)S_0$ where $\epsilon\ne r$. This is similar to a risk-neutral outlook except that, well, it is no longer risk-neutral.

Since $V_1$ and $S_1$ depend on head vs. tail, we have $V_1=\alpha S_1+\beta$ for some constants $\alpha$ and $\beta$ that can be calculated. The relationship between $V_1$ and $V_0$ has the property that
$$
E(V_1/V_0)=E(V_1)/V_0=\frac{\alpha E(S_1)+\beta}{\alpha S_0+\frac{\beta}{1+r}}
$$
since a bond is still risk-free hence risk-neutral when the interest rate is known.
$$
=\frac{\alpha (1+\epsilon)S_0 +\beta}{\alpha(1+r)S_0+\beta}\cdot (1+r)
$$
When $\epsilon=r$ this simplifies to $1+r$; in particular then $E(V_1/V_0)$ is the same for all securities $V$ when $\epsilon=r$. For any value of $\epsilon\ne r$, $E(V_1/V_0)$ depends on $\alpha$ and $\beta$.

Can we still find $V_0$ if $\epsilon\ne r$? Yes, the value of $V_0=\alpha S_0+\frac{\beta}{1+r}$ does not depend on $\epsilon$. But using $\epsilon\ne r$, we cannot relate $V_0$ to $E(V_1)$ in a simple way.

Updated September 24.

Each measure has the property that if for a random variable $X$, $\tilde E(X)=0$, and $\tilde P(X>0)>0$, then $\tilde P(X < 0)>0$.

In the study of interest rates, the so-called risk-neutral measure can to some extent be simply a fixed measure giving positive probability to each outcome.
Such a measure of course has the property that if for a random variable $X$, $\tilde E(X)=0$, and $X(\omega)>0$ for some $\omega\in\Omega$,
then $X(\omega) < 0$ for some $\omega\in\Omega$.

Now let us consider another measure $P$ which is the “real” probability, and which also gives positive probability to each outcome.

Then we have that if for a random variable $X$, $\tilde E(X)=0$, and $P(X>0)>0$, then $P(X < 0)>0$.
So if $X$ is the outcome of some trading strategy, then if we can verify that $\tilde E(X)=0$ then we will have ruled out arbitrage.
Now we can use $\tilde P$ and $\tilde E$ to define the “fair” prices of assets, in such a way that $\tilde E(X)=0$ follows.
Then these prices will indeed be fair, or at least arbitrage-free.

It does not follow that $\tilde P$ is in any sense “risk-neutral”, however. In Chapters 1–5 we had a stock valued at $S_0$ and moving to a future value $S_1(\omega_1)$. Then we defined the risk-neutral measure by the axiom $(1+r)S_0=\tilde E(S_1)$ (the equation that expresses risk-neutrality). That made sense because any other price $\hat S_0$ for the stock at time 0 would lead to an arbitrage: the stock would be trading at $S_0$ but would really be worth $\hat S_0$! (It is as if you could buy a 5-dollar bill for 1 dollar.) But in Chapter 6, there is no asset whose price at time 0 is given, there are only the random interest rates, and so we just fix one measure $\tilde P$ to price with.

That being said, there could well be an ambient stock to determine a risk-neutral measure:
Namely, the interest rate $R_n$ applied during $[n,n+1]$ is already known at time $n$, so it is possible to determine a risk-neutral measure from the stock price paths. Conversely, from the strange risk-neutral measure given in the first example of Ch. 6, and from the interest rates added to that example, one can determine a set of stock price paths (Figure 6.3.1). This procedure is not unique, though. For instance,
$$
\frac{1+r-d}{u-d}=\tilde p=P(HHH | HH) = 2/3
$$
and $\tilde q = P(HHT | HH) = 1/3$. And the interest rate is $r=R_2(HH)=1$. This gives
$$
\frac{2-d}{u-d}=\tilde p=P(HHH) = 2/3
$$
$$
3(2-d)=2(u-d)
$$
$$
6=2u+d
$$
For instance $u=8/3$ and $d=2/3$.

And $P(HTH | HT) = 1/2$ with $R_2(HT)=0$, giving
$$
\frac{1+0-d}{u-d}=\tilde p=1/2
$$
$$
2(1-d) = u-d
$$
$$
u+d=2
$$
For instance, $u=3/2$ and $d=1/2$.

Conclusion: it does makes sense to call $\tilde P$ a risk-neutral measure in Ch. 6 on interest rates, even though its risk-neutrality is not necessary for its use in that chapter.