Alternative proofs of the Krylov-Bogolioubov theorem

Let me first mention that actually Bogolyubov himself proved "the Krylov-Bogolyubov" theorem for actions of arbitrary amenable groups in 1939 (2 years after the Krylov-Bogolyubov theorem was published). Unfortunately, it was published in Ukrainian in a rather obscure place, and remained virtually unknown (see Anosov's article MR1311227 in Russian Math. Surveys 1994 for more details). Currently its Russian translation is available in various volumes of Bogolyubov's collected works, but as far as I know no English translation exists. Bogolyubov's argument is close to your "first proof", except for he works directly with means instead of Følner sequences (which had not been defined yet at the time).

Now, returning to your question about the "third proof". One way to do it (with a bit of cheating) consists in trying to replace invariance with respect to all elements of an amenable group with invariance just with respect to a single operator. This can, indeed, be done by using the fact that for any amenable group there is a probability measure $\mu$ on the group such that the sequence of its convolutions strongly converges to invariance (see the answer to the question characterizations of amenable groups which use the space $\ell_1(G)$ and convolution). Therefore, if a measure on the action space is invariant with respect to the convolution operator $P_\mu$ associated with the measure $\mu$, then it is necessarily group invariant. Now, for proving existence of $P_\mu$-invariant measures one can apply your argument. Of course, the same trick works for the "first" and for the "second" proofs as well.

PS The term "amenable action" already has several well established (and contradictory) meanings, and it is misleading to use it just for actions of amenable groups as you do in the last paragraph of your question.

Your proof can be modified a bit so that it works for a general countable, amenable group $\Gamma$. In this case, we can take $B(X)$ to be the closure of $\mbox{span} \{ f \circ T_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ and we show that $B(X)$ doesn't contain the constant functions on $X$ as follows:

I claim that every function $h \in \mbox{span} \{ f \circ T_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ is non-negative at some point in $X$, namely $\max_{x \in X} h(x) \ge 0$. This is sometimes called the "translate property" for amenable groups, if I'm not mistaken. Anyway, after establishing that, the conclusion follows easily: it follows that one cannot uniformly approximate a negative constant (say $-7$) by a function of the above form.

The proof of the translate property I'm familiar with goes like this:

Assume for contradiction that $\max_{x \in X} h(x) = \delta < 0$ and let $\varepsilon > 0$. Let $h=\sum_{i=1}^{n}f_{i}\circ T_{\gamma_{i}}-f_{i}$. Since $\Gamma$ is amenable we can find a finite set $U \subset \Gamma$ such that $\frac{\left|\gamma_{i}U\triangle U\right|}{\left|U\right|}<\varepsilon$ for all $i=1,\dots,n$ (the Følner condition). Now consider the function $F:=\frac{1}{\left|U\right|}\sum_{u\in U}h\circ T_{u}$. Choose some point $x_0 \in X$. On the one hand, by our assumption, $F\left(x_{0}\right) = \frac{1}{\left|U\right|}\sum_{u\in U} h \left(T_{u}x_{0}\right)\le\delta<0$. On the other hand,

$\left|F\left(x_{0}\right)\right|=\left|\frac{1}{\left|U\right|}\sum_{u\in U}\sum_{i=1}^{n}f_{i}\left(T_{\gamma_{i}}T_{u}x_{0}\right)-f_{i}\left(T_{u}x_{0}\right)\right|$

By the triangle inequality, changing order of summation and using the property of U, you see that the above is at most $2n \varepsilon \max_{x\in X,1\le i\le n}\left|f_{i}\left(x\right)\right|$. So if you choose $\varepsilon$ small enough, you get a contradiction.

Please check Krasnosel'skii, M.A.; Lifshits, Je.A.; Sobolev, A.V. (1990), Positive Linear Systems: The method of positive operators, Sigma Series in Applied Mathematics,

Krasnoselskii told me that he presented his argument at Bogolubov's seminar in 1949?