While early microcomputers were often 4- or 8-bit designs, with larger word sizes coming later, that pattern did not hold for their predecessors. None of the earliest computers used a word size that was anywhere near that small. Even computers which performed addition one bit at a time (e.g. Anastov's engine) still grouped bits into rather large words.
I suspect part of the reason machines used to use such large words was that the word size used to impose an upper limit on the largest number a machine could process without a huge (greater than 4x) loss in efficiency. On a machine with a 16-bit word size, adding together 1000 numbers which could each be anywhere from -32767 to +32767 would require a huge amount of work for each number. If the machine didn't have an efficient right-shift operator, the computation would likely require 2-3 conditional branches for each value to be added. Hugely expensive.
On microcomputers, however, there is usually a "carry" flag, and in many cases there will also be "add-with-carry" instructions which are no more expensive than "ordinary" add instructions [in some cases, including the 4004, addition without carry is more expensive than addition with]. Thus, while a 32-bit computer would have been limited to performing 32-bit math efficiently, an 8-bit microprocessor could perform math efficiently on 8, 16, 24, 32 40, 48, etc. bit quantities with a cost proportional to the integer width (for addition or subtraction) or the product of the source operands' widths (for multiplication). Since most operations won't need 32-bit quantities, using a smaller word size reduces hardware requirements without impacting efficiency of the dominant use cases, while still keeping the ability to perform longer operations when needed.
When did the carry flag originate? The Wiki page on the carry flag indicates it was present on the 4004 (apparent from its instruction set), but that hardly implies it was the first machine to use one. Did other kinds of computers start supporting multi-word arithmetic before the 4004?