The sense that my training is part of all the Aikido training that is going on in the world, a sense of connectivity to the roots of Aikido, and the right to drop by the Hombu if I am in town without having to work connections to get an introduction, etc.
My hunch would be that dropping in at hombu isn't high on the list for the average person. If they do, it doesn't take connections or introductions, just pay your money and go upstairs. Worst case is they make you pay the Aikikai entrance fee - about $100, but still less than testing fees.
What if your instructors weren't affiliated with the Aikikai? Would that make any difference at all to a sense of "connectivity"? When I started with Saotome he wasn't with the Aikikai, and all of my early ranks came directly from him, not from the Aikikai. As I recall, nobody cared, really, or felt less "connected" to the roots of Aikido, or felt more "connected" once he reaffiliated with the Aikikai (test fees went up a lot, though).