The studies that have been done primarily followed cases associated with heatstroke (not necessarily hospitalized) during the 1995 Chicago (almost a thousand incidents) and 2003 France heat waves (several thousand incidents). This means they all primarily have focused on classical heat-stroke vs exercise induced heat-stroke. It sounds like you've mostly worked with exercise induced heat-stroke, which as far as I can tell has ~5% death rate in the US: it kills you faster but is also far more survivable, likely because (1) interventions happen sooner and are more clear (more water, submersion/misting, cessation of activity, get out of the sun) and (2) the length of time overheated is far shorter. Classical (which basically means "externally induced" but which also includes side-effects of some drugs) seems harder to survive, likely because unlike exercise-induced its much harder to remove the heat source/exit the hot environment and length of time exposed to the heat tends to be much longer.
For the Chicago data I haven't found the totals that included non-hospitalized patients again (I'd read up on that larger pool study but forgot to bookmark it, 600 excess deaths were recorded for the heat-wave), but for those hospitalized (58 patients) it was basically 21% died in the hospital and another 28% died within the first year, with no patient having seen improvement in their functional status during that year. Also note the signs of infection these patients were admitted with does not necessarily indicate a pre-existing infection: heat-stroke's most deadly mechanism may actually be permeability changes leading to contents of your intestines and colon flooding the bloodstream.
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Near-fatal heat stroke during the 1995 heat wave in Chicago - PubMed
For the France study I similarly haven't found again the study that tried to look at the non-admitted data as well, but for the 345 admitted cases 62.6% died in hospital
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Mortality of patients with heatstroke admitted to intensive care units during the 2003 heat wave in France: a national multiple-center risk-factor study - PubMed
For just the town of Lyon, France there were 83 admitted cases of which 58% died in the first 28 days and 71% of which had died by 2 years, again patients showed little functional improvement over that 2 year time frame.
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Short- and long-term outcomes of heatstroke following the 2003 heat wave in Lyon, France - PubMed
Note: practices for cooling have improved and knowledge around the need to cool and cool quickly has improved since these things occurred, but I would still be surprised if for cases that did not have rapid interventions if the stats have moved much from this. From everything I've read on how heat-stroke operates, especially for classical heat-stroke (which is likely what ultimately killed Phil... a thing I'll explain at another time) if it's not treated quickly you're left in such bad shape it's unlikely you'll have much of a life if you live.