No it doesn't actually explain the Trinity. The trinity says that the son the father and the holy spirit are all God; but the son is not the father, the son is not the holy spirit and the father is not the holy spirit. By contrast, the elements in your illustration are all mutually equivalent, meaning in addition to the father son and husband all being a man; the father is also a son and a husband. See the difference? Your illustration is a good analog for modalism - not the trinity.
A better illustration would consist of 3 different persons that are all members of the same family: a set of triplets name Farter, Sean and Holigose, all surnamed Goad.
Farter is Goad,
Sean is Goad,
Holigose is Goad.
Farter is not Sean
Farter is not Holigose
Holigose is not Sean.
The Bible does not explicitly teach a trinity, but neither is there anything in the Bible that explicitly disproves the orthodox trinity teaching. However, the Bible does indeed show Jesus being given worshipful honor - something JWs loathe. I think it's pointless arguing the Trinity with JWs. A better approach is to use the Bible to show JWs that early Christians honored Jesus in a worshipful way that JWs today aren't allowed to honor him. I think that alone is enough to show them that their version of Jesus does not line up accurately with scripture and that their organization, by discouraging - forbidding - the worshipful honor of Jesus, is revealing itself as bordering on antichrist if not outright antichrist.
JWs distort the trinity doctrine into a modalism strawman and argue against modalism while foolishly thinking they're disproving the trinity. The trinity doctrine is as unfalsifiable as the uncreated, eternal, invisible god doctrine. Anyone who criticizes the trinity doctrine as illogical must also criticize the uncreated, eternal, invisible god doctrine as illogical, or resign to the fact that he's making himself a hypocrite by cherry-picking which illogical belief he'll accept and which he'll refuse.