= Solution
Write $a$ for the scalar intercept denoted $\mu$ in the question, to distinguish it from the individual means $\mu_i$. The <log-likelihood> of the <independent> <Poisson observations> is
$$
\ell(a,\beta)=\sum_i\left[y_i(a+\beta x_i)-e^{a+\beta x_i}-\log(y_i!)\right].
$$
Differentiating gives the <score equations> for an interior <maximum-likelihood estimator>:
$$
\boxed{\sum_i(y_i-\widehat\mu_i)=0,\qquad
\sum_i x_i(y_i-\widehat\mu_i)=0,\qquad
\widehat\mu_i=e^{\widehat a+\widehat\beta x_i}.}
$$
In particular, when $m=\sum_i y_i>0$,
$$
e^{\widehat a}=\frac{m}{\sum_i e^{\widehat\beta x_i}},\qquad
\frac{\sum_i x_i e^{\widehat\beta x_i}}{\sum_i e^{\widehat\beta x_i}}
=\frac{\sum_i x_i y_i}{m}.
$$
The left side is a weighted covariate mean whose derivative is its weighted <variance>, so it is strictly increasing when the covariates are not all equal. An interior solution is unique if the observed count-weighted mean is strictly between the smallest and largest covariates. At an extreme it may require a limiting slope, and if all counts are zero the maximum occurs as $a\to-\infty$, rather than at finite parameters. Constant covariates make the slope and intercept unidentifiable.
Put $S_j=\sum_i x_i^j\mu_i$. The negative <Hessian matrix> is deterministic at given parameters and is also the <Fisher information matrix>:
$$
I(a,\beta)=\begin{pmatrix}S_0&S_1\\ S_1&S_2\end{pmatrix},\qquad
I(a,\beta)^{-1}=\frac1{S_0S_2-S_1^2}
\begin{pmatrix}S_2&-S_1\\-S_1&S_0\end{pmatrix}.
$$
Under the usual interior, increasing-information regularity for <asymptotic normality of a maximum likelihood estimator>, replace the unknown means by their fitted values and read the lower-right entry:
$$
\boxed{\operatorname{Var}(\widehat\beta)\approx
\frac{\sum_i\widehat\mu_i}{(\sum_i\widehat\mu_i)(\sum_i x_i^2\widehat\mu_i)-(\sum_i x_i\widehat\mu_i)^2}.}
$$
Equivalently, <Poisson slope information after eliminating an intercept> is $\sum_i\mu_i(x_i-\bar x_\mu)^2$, with $\bar x_\mu=S_1/S_0$. This explains both the positive denominator for varying covariates and the loss of slope information when their spread is small.
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