= Solution
There is a genuine flux typo in the original PDF. A received bolometric <radiative flux> must have inverse-square <luminosity distance>, so the physically consistent <thermal bremsstrahlung> expression is $S_X\propto d_L^{-2}\int n_e^2T_e^{1/2}\,dV$. The printed single power of $d_L$ cannot imply the requested <Hubble constant> scaling.
For an approximately uniform, spherical <galaxy cluster> let its depth be $\ell=q\theta d_A$, where $q$ is a fixed geometrical factor. With fixed profile coefficients $A,B$, the <Rayleigh-Jeans law> limit of the <thermal Sunyaev-Zeldovich effect> and the bolometric X-ray <radiative flux> give
$$
D\equiv|\Delta T_\gamma|=A n_eT_e\ell,\qquad
S_X=B\frac{n_e^2T_e^{1/2}\ell^3}{d_L^2}.
$$
Eliminate the electron <number density> and use $d_L=(1+z)^2d_A$:
$$
S_X=\frac B{A^2}\frac{D^2\ell}{T_e^{3/2}d_L^2}
=\frac{Bq}{A^2}\frac{D^2\theta}{T_e^{3/2}(1+z)^4d_A}.
$$
The <cluster distance from Sunyaev-Zeldovich and X-ray signals> is consequently
$$
\boxed{d_A=\frac{Bq}{A^2}\frac{D^2\theta}{S_XT_e^{3/2}(1+z)^4}.}
$$
At low <cosmological redshift>, $d_A\simeq cz/H_0$, so
$$
\boxed{H_0\simeq\frac{c z A^2}{Bq}\frac{S_XT_e^{3/2}(1+z)^4}{D^2\theta}
\ \propto\ (\Delta T_\gamma)^{-2}}
$$
when the other observables and profile parameters are held fixed. With the PDF's literal $d_L^{-1}$ instead, $S_X/D^2\propto\ell/(T_e^{3/2}d_L)=q\theta/[T_e^{3/2}(1+z)^2]$, independent of distance: this explicitly shows why the correction is necessary.
Real <galaxy clusters> are neither uniform nor generally spherical. The inferred depth depends on the ratio of <line of sight> depth to transverse extent; elongation and orientation selection therefore bias the distance. <density> clumps enhance X-ray emission, which weights $n_e^2$, more than the <thermal Sunyaev-Zeldovich effect>, which weights $n_e$. An unmodelled <gas clumping factor> greater than one makes the inferred distance too small and the inferred <Hubble constant> too large in this simple model. <Temperature> gradients and multiple thermal components bias the X-ray spectral temperature relative to the pressure-weighted <Compton y parameter>. Metals alter the cooling spectrum, and finite X-ray bands require a spectral and redshift correction to the bolometric approximation. Relativistic thermal corrections, the <kinetic Sunyaev-Zeldovich effect>, radio sources and primary <Cosmic microwave background anisotropy> contaminate the signal. Finally, calibration, angular-profile fitting and low-redshift <peculiar velocities> affect the observables and the relation between measured redshift and expansion distance.
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