For a definite answer, you'd have to ask Stephen Morse, the designer of the 8086 instruction-set.
Other Intel engineers worked on the actual implementation, but apparently the ISA was designed on paper first, almost entirely by just one guy. He's also credited as principal architect of 8086. PC World interviewed him in 2008, for the 30th anniversary of 8086, and more importantly he wrote a book, The 8086/8088 Primer (1982). I haven't read it, but apparently he discusses some design decisions as well as how to program it. If you're lucky, maybe he wrote something about choosing to have div/idiv trap.
There's no reason it had to be this way; setting CF and/or OF and truncating would have been valid designs. But you still need to choose some value to put in the quotient/remainder output registers in the divide-by-zero case1. (I think it's fairly common for ISAs with HW division to have a divide-by-error exception for at least divide by zero, but On which platforms does integer divide by zero trigger a floating point exception? unfortunately only mentions x86 as an ISA with traps. If division does trap and a POSIX OS delivers a signal at all, it must be SIGFPE for an arithmetic exception.)
Note that other ISAs in do make different choices. For example ARM division never faults, and doesn't set flags either. (Although it doesn't provide a double-width dividend, so only the INT_MIN / -1 signed overflow and division by 0 cases are special for it.)
IDK if building a hardware divide unit (or microcode) that could get the correctly-truncated quotient for overflow cases (when the exact quotient is wider than 16-bit) would be harder than simply detecting overflow and bailing out. If so, that would be a fairly good reason.
(Leaving garbage in the output registers and setting FLAGS would be possible but not great; every division would need to check the result afterwards if it wanted to avoid the possibility of using garbage.)
Note 1: div by 0 in some ways is a special case of this: high_half < divisor is false for divisor=0 for any dividend. But there's no well-defined mathematical result to truncate. IEEE FP division resolves this by treating as the limit as divisor approaches 0, i.e. +- infinity. But integer 0 should be assumed to be exactly 0, not some tiny number, and there's no in-band NaN or Inf value to use anyway, only a finite 0xFFFF...
No other 8086 math instructions need to truncate, if you consider FLAGS
Note that 8086 only included the one-operand forms of mul and imul, which do widening multiply: DX:AX = AX * src. (CF and OF are set if the high half is non-zero (for mul), or if the high half isn't the sign-extension of the low half (for imul)). Only later CPUs introduced truncating forms like imul r, r/m, imm (186) and imul r, r/m (386) that don't waste time writing the high half anywhere, although still setting FLAGS so you could detect signed wrapping if you wanted. (Most uses don't, so later CPUs only provided imul, versions of mul that would be the same except for FLAGS.)
add/sub can carry/borrow, but the full result of an add is available as CF : reg with the extra bit in the carry flag.
If you consider sar / shr / shl reg, cl as a bitwise logical operation, not math, then it doesn't count even though it can shift out multiple bits without leaving them anywhere. (The last bit is left in CF, so shift-by-1 can be undone with a rotate-through-carry.)
That leaves DIV / IDIV as I think the only arithmetic instructions where there could be a wider result and nowhere to put it. That might have been part of the motivation for choosing to have them fault.
high_half < divisor is gotcha-proof for unsigned division
That's the exact condition for the quotient fitting in the operand-size. 1:0 (e.g. 0x0100 for 8-bit operand-size) is smallest quotient that doesn't fit, so 0x0100 * divisor is the smallest dividend that would produce a quotient that doesn't fit in 8 bits.
That dividend is divisor:0 when split up into hi:lo halves of the same width as the dividend.
Any number smaller than that would have to "borrow" from the high half, leaving it strictly smaller than divisor.
(Signed division also has the INT_MIN / -1 overflow corner case, and the high-half check might have to be on absolute values.)
