Home advantage as policy
The rule that the lower-division club hosts is the single most consequential piece of cup design. It moves the tie to a smaller ground with a partisan crowd, an unfamiliar surface and a dressing room the favourite has never used, and it guarantees the smaller club the matchday income that makes a cup run financially transformative. What looks like a courtesy is in fact the main reason the competition produces the stories it does.
It is also a reasonable answer to a fairness question. A neutral draw would send most small clubs to large stadiums, where they would be beaten more comfortably and earn a share of somebody else's gate.
Reserved places for the amateur game
Allocating entry places to regional cup winners means the competition is genuinely open to clubs several divisions down, and it gives the regional competitions a prize worth chasing. A season of local cup football is transformed by the prospect of drawing a famous opponent, and the clubs involved plan and invest around it in ways that strengthen the wider game.
The cost is a handful of one-sided early ties. Set against a functioning pyramid where any club can dream of the draw, that seems a small price.
Seeding the early rounds
Separating top-flight clubs in the opening rounds is often criticised as protecting the powerful, and in a narrow sense it does. The intent, though, is to preserve the competition's shape: if half the top division were eliminated by each other in the first round, the later rounds would offer fewer of the mismatches that make the tournament distinctive, and the broadcast value that funds the prize money would fall.
Every cup makes this trade between purity and design. Being explicit about it is better than pretending a draw is neutral when the rules plainly steer it.