Adding HCl would indeed make it more potent, but it would also likely destroy/dissolve the metal (particularly stainless) of the ice machines you are cleaning.
Pure phosphoric acid is already a sticky, syrupy mess! I doubt it would work better than a dilute solution. It is perhaps counterintuitive that highly concentrated solutions do not always work best.
You could always try thickening with carbomer. It should be stable to something tame like phosphoric acid.