I wouldn't bother too much about the logical consequences of arcane spell failure. The whole concept of arcane spell failure isn't very logical in the first place and the rules don't even give a good ingame explanation for it. Divine spells do use somatic components too, but a divine spellcaster does never risk losing a spell, even when wearing Mountain Plate and wielding a Tower Shield. The PHP states: "Wizards and sorcerers do not know how to wear armor effectively", but this reason can be disproved all too easy. A Wizard/Cleric/Mystic Theurge does indeed know how to wear armor effectively, and can cast divine spells even if armor-clad from head to toe, but arcane spells will constantly fizzle.
So what purpose does arcane spell failure serve? In my opinion there are three:
1.) It is a relic from previous D&D Editions, where Wizards couldn't cast when wearing armor at all.
2.) It fits the role model of an arcane spellcaster (Wizards and Sorcerers aren't supposed to wear heavy armor, but are supposed to be physically weak and fragile).
3.) It is a balancing issue. Armor proficiency can be obtained all to easy in D&D 3, overcoming spell failure is much trickier.
So what does this mean for the original case? Simple ignore the cane-on-a-rope. It isn't logocal that a Wiz has always the required spell component handy in an instant from his component pouch, although a Wiz has to haul around a lot of components. To be honest, this is downright absurd, but the rules explicitly state that it is supposed to work. The same applies to the cane.