I dug up an old Open Phil newsletter that you might be interested in.
It's ~4-5 years old so some things might have changed since then, but it addresses some of the questions you pose here quite directly.
I'm not sure it would change your underlying conclusions, but for what it's worth I took a quick look at the data/methods from Human Climate Horizons and think that it is likely an underestimate of future heat-related mortality.
This paper underlies the Human Climate Horizons/OWID estimates. A few quick assumptions I think are worth highlighting:
1. The authors (by admission) do not consider the effects of humidity. Accurate humidity data is much harder to come by than temperature, and it's often not included in the down-scaled versions of the climate models, though some exceptions apply. However, it's potentially quite relevant for heat-related mortality, though there is some debate in the literature. Given that it is unlikely that incorporating humidity would decrease heat-related mortality, my own view here is that this pushes current estimates towards a lower bound.
2. The authors use climate model data that are downscaled with a technique called Bias-Correction Spatial Disaggregation (BCSD). This approach has two key assumptions in it: (1) it assumes that the relative spatial patterns in the training data will remain constant under future climate change, and (2) it is calibrated to monthly data, not day to day changes, which has the effect of dampening extreme values. In practice, these assumptions limit the ability to model things like extreme heat waves and heat domes, which can cause large fatality spikes (e.g. figure below from Washington State in 2021). Missing these features in some locations might be akin to missing almost all the possible heat related mortality in cooler climates.Again, I don't think these are significant enough to fundamentally change your conclusions, but I do think it's worth highlighting that these types of results can be quite sensitive to specifics of the climate modeling approaches that are used.
I had several conversations with 'specialists' after my 80k advising call, and personally found the seemingly generic 'tell me about your career path' information surprisingly helpful! I found that people usually ended up sharing little tidbits of information that helped clarify the landscape of different fields, who is entering them and by what channels, and what alternative pathways might look like.