DRU(D)GE DToRaK[H] Tet-Resh-Het
TOR-AKH_______טרח_______[DT-R-KH à DRG]
ROOTS: There is no Indo-European (IE) "root" for DRUDGE. The origin of DRUDGE and DRUDGERY is listed as "uncertain," but Anglo-Saxon dreogan is a person who does hard or unpleasant work. Old Irish drugaire is a slave. Middle English druggin is to work hard.
The second D in DRUDGE is non-historic.
The D does appear in Middle Hebrew טרדה DTiRDaH (bother), from an Aramaic word of "troublesome" and the Akkadian meaning "pursued."
In Modern Hebrew טרוד DTaRooD means the less harried "busy, occupied."
The oldest root should echo DRG, not DRD.
טרח DToRaK[H] is "labor" or "endeavor," translated "cumbrance" (now encumbrance) or "trouble" in Deuteronomy 1:12. A wearisome burden is the sense in Isaiah 1:14; the verb הטריח HeeDTReeYa[K]H (he burdened) is in Job 37:11. In Aramaic the verb means to take pains or trouble.
Closer to the English noun DRUDGE, a טרחן DTiRK[H]aN in Modern Hebrew is a bothersome person or noodnik.
The work part of a process is the דרך DeReKH, the path or way (see "DIRECTION.")
BRANCHES: INTRICATE and the slang DRAG (a boring chore or any wearying experience) should be related. DRUDGERY in Dutch is tööorjus. Lithuanian and Polish drudgery is katorga.
Trud in Polish means trouble, toil, etc. Similarly, Romanian trudă means toil, labor, work, drudgery or trouble.
Russian "toil" is трудиться trudit'sa. Scottish drug is "a rough pull," a real DRAG that seems to DRAG in the pulling DRAG. DRAG is defined by Webster's as "to continue tediously," then "prolonged tediously."
The painful sense of טרח DToRaKH appears to be more important than an act of hauling. Reducing this DT-R-[K]H labor/laborious root to DR, there is an IE "root" dere (to work) which takes in DRASTIC and DRAMA. While the pulling form of DRAG is laborious, it is not clear if it or דרך DaRaKH (to tread) is more than distantly related. TRUDGE (no IE "root") is a wearying, laborious walk, so it should belong here with both דרך DaRaKH and טרח DToRaK[H].
Besides the DRG Irish slave above, Old Irish trogae, misery, is likely expressing the strongly painful aspect of טרח DToRaK[H].
More hard work at "AMERICA" and ROBOT."