Solved – what problems can occur when substituting zeros for gamma regression

The description "common practice" may fit literature you know, but I've not seen this often otherwise. Similarly, "randomness" is the wrong word here: "arbitrariness" perhaps fits the case better.

Note that the replacement number, even if you favour this approach, should certainly not be any value that might genuinely occur otherwise, as is likely for 1. Furthermore, thinking in terms of "a very small number" is the wrong way to think, as its logarithm would be large negative and that could create massive outliers on that scale.

If there are substantive grounds for regarding zeros as really small positives, some adjustment might make sense.

But the fudge should be unnecessary here. Even if you use a log link with the gamma family in GLM the key point is not whether there are zeros in the data (so that log of zero makes no sense), but the assumption that means are positive. If you use some other link, the same key point should solve the problem.

The comments here naturally do not rule out other checks, e.g. sensitivity checks to possible adjustment of recorded zeros, or indeed other models should they seem appropriate (zero-inflated gammas, or whatever).