Perhaps what should be guiding your decision is whether the sample you'll be working with meets the definition of zero truncated sample - i.e.,

A dataset in which observations enter the sample only after the first count occurs.

For instance, the number of times a person seeks medical treatment at a hospital is a zero truncated sample if the observations are based on admitted patients. But in your case, dealing with treatment that might "revert" a health condition seems to violate the definition of zero truncated sample, I think. Actually, the case you describe seems adjacent to applications of Survival Analysis as those described in this textbook, but I will refrain from going off on a tangent.

There are alternative viewpoints when it comes to whether or not one should zero truncated counts: for instance, with reference to the zero-truncated Poisson regression in ch.7 of this textbook Hilbe claims that when the count response variable has a mean greater than 5, we simply should not expect many zero counts in a Poisson model---possibly less than 1% in any given dataset. This type of reasoning would favor a more "relaxed" approach, but it refers to a different specification.

With specific regards to the Negative Binomial - whether or not left-truncated at zero - one typically expects that the choice be driven by aspects of the data being investigated, such as for instance over-dispersion. You don't seem to elaborate on that in your query - hence I thought I'd bring it up. Classic textbooks from Hilbe and Cameron and Trivedi provide guidance regarding how to test your data.

As a corollary - model selection can be tricky. In Fig.3 of this academic paper you'll find a bit of a "workflow" on how to select the specification that might work best for you - even if the context is management research.

Last but not least: from a practical/computational perspective the zero-truncated Negative Binomial is just trickier to implement under the hood - as you can see in this post it's giving me some headache. To keep it practical: zero-truncation would go beyond the capabilities of widely-used glm fitting procedures in R such as glm.nb from the package MASS. So much so that the same textbook from Hilbe I mentioned earlier reverts to other packages such as GAMLSS which operate on a different statistical framework than GLM.

To close off - this classic paper covers most of the essential aspects of zero truncated Poisson and Negative Binomial regression, I wonder if it could provide some guidance.

I hope the above is somewhat useful.