= Solution
An <ignorable missingness mechanism> need not be absent from the data-generating process. It means that <likelihood> inference about the data-model parameters can omit the missingness factor, while still integrating over missing values.
Write the complete joint data <statistical probability density> as $f_\theta(y,x)$ and the conditional missingness <probability> as $g_\psi(r\mid y,x)$. All $y$ values and only $x_{\mathrm{obs}}$ are observed. Under <missing at random>, $g_\psi(r\mid y,x)$ is constant as $x_{\mathrm{mis}}$ varies with $(y,x_{\mathrm{obs}})$ fixed. The actual <observed-data likelihood> is therefore
$$
\begin{aligned}
L(\theta,\psi;y,x_{\mathrm{obs}},r)
&=\int f_\theta(y,x_{\mathrm{obs}},x_{\mathrm{mis}})g_\psi(r\mid y,x_{\mathrm{obs}},x_{\mathrm{mis}})\,dx_{\mathrm{mis}}\\
&=g_\psi(r\mid y,x_{\mathrm{obs}})\underbrace{\int f_\theta(y,x_{\mathrm{obs}},x_{\mathrm{mis}})\,dx_{\mathrm{mis}}}_{L_{\mathrm{obs}}(\theta;y,x_{\mathrm{obs}})}.
\end{aligned}
$$
The assumed distinctness is understood as independent variation of $\theta$ and $\psi$. Maximizing over $\psi$ multiplies $L_{\mathrm{obs}}$ by a factor independent of $\theta$; <likelihood ratios>, scores and <likelihood> curvature for $\theta$ are therefore unchanged by omitting $g_\psi$. This proves <likelihood> ignorability.
For implementation of the <observed-data likelihood with a missing covariate>, a <linear regression> model for $Y\mid X$ must be accompanied by an appropriate model for the distribution of $X$. For independent individuals, write $f_{\beta}(y\mid x)$ for the regression <statistical probability density> and $g_\eta(x)$ for the age <statistical probability density>. Up to the ignorable factor,
$$
\boxed{L_{\mathrm{obs}}(\beta,\eta)=\prod_{i:R_i=1}f_\beta(y_i\mid x_i)g_\eta(x_i)\ \prod_{i:R_i=0}\int f_\beta(y_i\mid x)g_\eta(x)\,dx.}
$$
Ignorability does not authorize discarding missing-age cases or assuming their ages have the distribution seen in the complete cases. It removes the need to model the observation mechanism for <likelihood> inference under the stated conditions, not the need to handle the missing covariates.
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