The role of H$_2$ is to allow primordial gas to cool down sufficiently to allow the collapse to start and then to hold the gas at a relatively low temperature as it gets much more dense. The formation of H$_2$ is essential because atomic hydrogen simply has no way of cooling itself below temperatures that are capable of exciting the $n=2$ level and then emitting Lyman alpha photons. This means that without H$_2$ the gas can't cool effectively once it gets below about $10^4$ K and it will never reach the densities required to form stars.
Without molecular H$_2$ you get stuck with large ($\sim 10^4 -10^6M_{\odot}$) pressure-supported "protogalaxies" that won't collapse any further. These are formed at redshifts of around 20-40 from growing primordial density perturbations and have initial temperatures of a few thousand K. The ongoing formation of H$_2$ in primordial gas phase reactions (no dust required) allows a collapsing cloud to cool down to around $\sim 200$ K.
The importance of H$_2$ cooling is outlined in reviews by Glover 2005 and reiterated more recently by Klessen 2018. A telling quote is from the review by Bromm 2013, who says
If the gas were unable to cool, there would be no further collapse, and consequently no
gas fragmentation and star formation. The gas would simply persist in hydrostatic
equilibrium, roughly tracing the density profile of the Dark Matter. Early on, it was realized
that cooling in the low-temperature primordial gas had to rely on molecular hydrogen
(H2) instead (Saslaw and Zipoy 1967).
Conversely, in gas that is enriched with heavier elements, H$_2$ cooling is totally unimportant because there are more efficient routes for a collapsing cloud to radiate away energy (e.g. Glover & Clark 2012).
