Re: 8088 & 8086
The 8086 was indeed a "proper" 16-bit processor. And internally, so was the 8088. The only difference between them was that the 8088 had an 8-bit data bus. That meant that it only moved 8 bits in and out at once. That allowed it move interoperate with common 8-bit hardware at the time—which was the real reason Intel made it, and why IBM chose it over the 8086. Newer 16-bit peripherals were not only more costly, but scarcer.
Your objection over the segmented architecture is orthogonal. The segments were 64 kB precisely **because** of the 16-bit registers. With 16 bits, you can only count from zero to 65,535, so you can directly address 65,536 bytes, commonly abbreviated using pseudo-SI units as 64 kilobytes (Wikipedia wants us to use the word "kibibytes", but that sounds like a brand of dog food).
Using segments was a clever way to allow the processor to address **more than** 64 kB. The segments served as a multiplier, adding 4 bits to the total address space, allowing 2^20 or 1 MB of addressable memory. Intel could have chosen a greater multiplier. In the 8086, the segments were shifted left by 4 bits, making them only 16 bytes apart from each other. If they had instead chosen to shift them by 8 bits, making them 256 bytes apart from each other, that would have theoretically given the CPU access to 16 MB of memory. And by shifting the segments by 16 bits and making them not overlap at all, the 8086/8 could theoretically have accessed 4 GB of memory. But just adding that many address lines to the CPU pinout would have been a greater complication, and expense, than could be justified by the computing needs at the time. This CPU was already a huge improvement over pretty much every other CPU available at the time, including the 8085 it succeeded, which could typically address only 64 kB **in total**.
The 286 was no more of a "real" 16-bit processor than the 8086/8 (or the 80186, for that matter). It still used 64 kB segments. But in protected mode, you had to set up a descriptor for each segment you used, which included, among other things, a 24-bit segment offset, giving the CPU access to up to 16 MB of memory.
The 386 still had segments and selectors like the 286 (as do its descendents today). But they're practically always set up to start at zero and then just ignored, since the address bus and register width are now the same size.