This is what fixed it:
Open the printer's ppd file /etc/cups/ppd/HP_LaserJet_IIIp.ppd.
Find the part with all the available options like paper size and resolution.
Insert this snippet:
*OpenUI *HalftoningAlgorithm/Halftoning Algorithm: PickOne
*OrderDependency: 1000 PageSetup *HalftoningAlgorithm
*DefaultHalftoningAlgorithm: Standard
*HalftoningAlgorithm Accurate/Accurate: "
<< /UseWTS false >> setuserparams
<<
/AccurateScreens true
/HalftoneType 1
/HalftoneName (Round Dot Screen)
/SpotFunction { 180 mul cos exch 180 mul cos add 2 div}
/Frequency 137
/Angle 37
>> sethalftone
"
*End
*HalftoningAlgorithm Standard/Standard: ""
*HalftoningAlgorithm WTS/Well-Tempered Screening: "
<< /UseWTS true >> setuserparams
<<
/AccurateScreens true
/HalftoneType 1
/HalftoneName (Round Dot Screen)
/SpotFunction { 180 mul cos exch 180 mul cos add 2 div}
/Frequency 137
/Angle 37
>> sethalftone
"
*End
*CloseUI: *HalftoningAlgorithm
(Source)
... and the menu item for setting the halftoning algorithm will show up again, both in applications and in the cups configuration itself, where the default can be set to "accurate".
Why it was removed from the ppd file in the first place, however, remains a mystery.