What I do believe, and what I mean when I say weapon based, or weapon context, is that weapons must be present OR POTENTIALLY PRESENT for the Aikido set-ups to become useful. That weapons must be present (along with multiple attacker situations) OR POTENTIALLY PRESENT in order to use the movement, distance and timing taught by the Aikido syllabus.
Meaning, for example, in the field of law enforcement, wherein everyone, from the gang-banger that's running upon seeing you to drunk granny out taking her evening walk, is assumed to be armed - where there is no tactical difference allotted for whether or not folks are armed. Everyone is treated as they are armed and all tactics follow that suit.
In that setting, the commonly practiced Aikido syllabus transfers more readily to combat than the commonly practiced MMA syllabus. The reason, in my opinion, is that weapons and/or the potential for weapons being present generates a space/time more fertile for Aikido waza to function and remain viable.
I think you've hit the nail on the head! The presents or potential presents of weapons changes the way someone will try to deal with you.
Context is king!
The context in which you are fighting makes all the difference. For example, if we look at the typical Aikido rush in and wrist grab attack, v.s. a wrestling double/single leg take down. If your context is unarmed (which is what most people first think of when they think "marital arts") the double/single leg take down seems like the clearly superior technique. Looking to MMA as an example, the double/single leg is a hugely successful technique, used regularly. While the "run in and grab katate" approach has seldom had much effect!
However let's picture a double/single leg done on a man with a holstered pistol. Very likely that the wrestler going for the double/single leg, on a man with a holstered pistol, will end up getting shot. The wrestler will have no control of the weapon hand, as soon as the armed man realized what is happening, he will draw and shoot. Contrast that to the "rush in an grab katate attack". When someone has a weapon, quickly getting control of his weapon hand now suddenly seems like a good idea. Attacking this way may allow us to get to the pistol before it is drawn, saving our life.
Context changes things dramatically, understanding what context our system fits into is of the utmost importance!