While I too am a huge fan of the legendary 68000, as well as the proud owner of many Amigas from 1985 onward, the marketing and media reports sometimes glossed over important technical details. The 68000 CPU, which all Amigas from 1985 to 1990 were designed around, does have 32-bit data and address registers but that doesn't mean it was purely a 32-bit architecture -
even internally. Some important internal components like the ALU were only 16-bit. Additionally, the external data width was 16-bit, requiring two accesses to read or write a 32-bit register to RAM, which did have a meaningful performance impact since memory access is a critical bottleneck, especially in a CPU with no cache. As you note, at least this 'double pumping' was automatic and mostly hidden from programmers.
The 68000's address registers didn't have their upper 8 bits connected to external pins, limiting the directly addressable RAM to 16MB (24-bits). These external width compromises allowed the 68000 to fit in a 64 pin DIP package while the standard 68020, which did connect all 32 data and address lines, came in a 114 pin PGA package. Large packages with more pins were a significant cost while double-pumping data accesses and a 16MB limit on addressable RAM weren't significant issues for most 1980s desktop computers - especially since the 68000's elegantly orthogonal instruction set was so performant in other ways.
Thus, many of us more technically literate fans broadly thought of the 68000 as having 32-bits internally but 16-bit data / 24-bit address width externally. However, that was incorrect because the arithmetic logic unit (ALU) and two arithmetic units were also 16-bit only, generally requiring at least twice as many cycles even for purely internal 32-bit math operations, whereas the 68020 and later CPUs didn't. That's why the 68000 is probably best described as "a hybrid 16/32 bit internal architecture with 16-bit external data width and 24-bit addressing."
It gets even more confusing because some later Amiga models like the A1200 (1992) didn't have a standard 68020 but instead a lower cost version, the 68EC020, which also didn't have the top 8 address lines connected and came in a smaller 100 pin QFP package. So technically, it had the same addressable RAM limit as the 68000, although it had full 32-bit internal and external data widths, ALU, a 256 byte cache and many other other instruction set and clock speed improvements common to later 680x0 CPUs. The way a lot of us thought of the 68000's 16/32 architecture as being limited just in the memory addressing was really a more appropriate description of the difference between a full 68020 and 68EC020. The 68000's ALU being 16-bit is the inarguable smoking gun that makes it incorrect to think "it's really a 32-bit CPU internally" as I used to.
However, that should take nothing away from just how incredible the 68000 was. My first computer had the 68000's little brother, the 6809, which was generally the fastest 8-bit CPU clock-for-clock due to being an 8/16 bit design in much same way the 68000 was 16/32 bit. While the 6809 was incredibly fast, when I got a 68000-based A1000 in 1985 and programmed it in assembly language, it blew my mind how incredibly fast it was. Then in 1988 when I added an A2620 accelerator card to my A2000, it's full 68020 with 32-bit internals and direct 32-bit read/write to 4MB of RAM was like going from a Ferrari to a Lear Jet. Despite how the 68000 was confusingly marketed and inaccurately described by some media, it was truly a leap forward, but the reality is the 68020 was really the first true 32-bit CPU in the line.