Why would you need any special case? If 1 means there is 1 item then you can still use 0 to mean there are 0 items.
The 1-based counting would be to index or label them like item 1, item 2, item 3, ... which is quite natural to humans. If you had zero items, you'd express that by your list being empty, not by having a number 0 somewhere. It's not a special case.
Lets consider a measuring tool for fluids which is based on volume. What is a useful range for the index on the side that measures the currently non-air capacity?
Many such devices, E.G. a measuring cup I've got at home in the kitchen, use a series of tick marks within a bounded scale of accuracy range ] with marks on both the top (of course) and also the bottom (it looks nice).
While 0 isn't expressly labeled on many scales, it is part of the inclusive range by implicit nature. Thus as you point out the case of 0 units, and 'empty set' are one and the same in this real world example.
I haven't done enough "fluid" applications to give domain-related suggestions. That's not really an integer "count". I'm only saying it doesn't fit well in typical business applications as I see them. Each domain is different.
Many of our existing conventions derived from military, science, and academic applications. Business applications came along later, emphasizing discrete counting and categorizing things along the lines of set theory, such as in the set or not in the set, not half in. (Money has decimals, but you don't typically increment through it with indexes.)
(Although COBOL was published around 1960, it took roughly 5 years for computers to get powerful enough to make it practical and widespread. The earliest COBOL compilers were dog-slow resource hogs relative to the hardware of the day.)
You forgot to explicitly mention that bottom mark on your measurement cup is indeed 0, and without it you could measure "at least this volume of fluid".
That doesn't change anything that I see. Indexes and counters are usually not used for "continuous" metrics such as liquid quantities. And again, the biz domain typically does not count fractional quantities, at least not in an indexed way.
It's not a practical problem for most coding. If you disagree, can you demonstrate a problem caused by indexing arrays starting at one in a typical code situation/scenario?
A computer should start counting from 0, there's no reason to special case the empty set.