You seem to see this with too modern eyes, as something demeaning. It's a mere mark of respect for the position, part of the protocol. A ritual.

Monarch don't owe obeisance to the Amyrlin Seat. No one but the Aes Sedai, the employees of the WT and the people of Tar Valon under her governance do.

The Amyrlin is considered to rank above the rulers in matters of protocol. There's absolutely no legal framework giving her any power over the rulers. She's nothing like an "High Queen" or an Empress as some mistakingly believe. No one but the sisters are tied to her with bonds of allegiance, the way the rulers under Fortuona are tied to the Empress. Perrin has theorically more power over Alliandre and Ghealdan than Egwene has over any nation but her own city-state.

The Amyrlin has no political power outside Tar Valon, she has influence - diplomatic and political, economic. Tar Valon oversees a major trade route, the Tower has the ear of several nations and has too much allies, and it has the One Power etc. The influence of the Amyrlin over nation is function of the perception of the people and their rulers of the weight and power of the Tower, and how much they value Aes Sedai. Andor sees Tar Valon has a major ally. The Bordermen in the east place high value on the Tower, especially Arafel and Shienar (not sp much Saldaea, if the opinions of Tenobia and Faile are any indication) . Far Madding can afford to be more independant, but for economical reasons won't dare cross Tar Valon either. In general, the nations of the South don't recognize the Tower as much influence as the nations of the north, which doesn't mean they'll risk making enemies of the Tower. It's really more a question of diplomacy than power.

What Elayne meant about the Tower and Andor is that she will draw a line between the sovereignty of Andor and her powers as Queen, and the power the Amyrlin has over her as Aes Sedai. She doesn't intend to let any Amyrlin extend powers she doesn't have over Andor using the excuse that its ruler is Aes Sedai.

I wouldn't fear for Elayne... right now she's the one who keeps crossing the line. In KOD for instance, she stripped the Black Ajah of their status as Aes Sedai by removing their rings. As she points out herself, Elayne acted impulsively about this and Egwene might be angry about what she's done because by Tower Law only the Amyrlin can strip a sister of her Great Serpent Ring. So Elayne usurped a prerogative of the Amyrlin. That's a triffle, merely an affront to the authority of Tar Valon Egwene might well let slip as long as Elayne is discreet about it.

Elayne crossed the line in TOM far more seriously, by securing the exclusive services of the Kin for her dual monarchy, admitting the Aes Sedai will retire in the Kin eventually etc. Elayne knows all too well that the Tower considers these women bound to Tar Valon and the Amyrlin has made known her plans for the group. Elayne's plans for the Kin should really have been negotiated between her and the Amyrlin and Hall, and it would have been a heated debate as in effect Elayne intends to violate the WT tradition of neutrality, by securing the services of a group associated to the Tower for her nation. This will spark resent and jealousy among the economical rivals of Andor. As Aes Sedai, Elayne had no authority to negotiate anything like this. The Hall is likely to be extremely angry with her. They already hold a grudge over the bargain with the Seafolk, though the main fault is Egwene's, who made the mistake of granting Elayne and Nynaeve authority, not foreseeing these two could go way too far and bound the Tower to something that should really have been have been negotiated between the Seafolk and the Hall of the Tower.

Elayne no doubt also violates Tower Law by freely giving ter'angreal to non-Aes Sedai. There's no law binding the nations to give their ter'angreal to the Tower, but diplomacy pushes them to comply with these wishes of the Tower. For AS, however, there's probably a law forbidding them from hiding or giving away Objects of the power. Elaida sure got AS flogged for hiding an angreal from the Hall.

The Hall would also have fits over the fact Elayne, openly Aes Sedai, has had women the Tower considers "wilders" (the Windfinders) use the OP openly as a weapon in her realm, by orders of her general.

So far it's not the Queen of Andor who has reasons to fear the Amyrlin might start interferring in Andoran affairs (well, Elayne did have reasons, as Elaida tried to do just that by ordering Elayne to name a regent - and that's the context in which Elayne said she wouldn't let her status as AS grant this sort of power to the Amyrlin over Andor), it's more the Amyrlin Seat and the Hall who have reasons to fear Elayne uses the fact she's Queen of Andor to overstep her power as Aes Sedai. Egwene always seemed to think Nynaeve was the "loose cannon" of the duo, and Elayne was merely influenced. It's far more Elayne Egwene should have worried about. It's Elayne who really needs a reminder where the authority of an Aes Sedai stops,, whether she's a friend of the Amyrlin or not, and where the Hall's and the Amyrlin's begin.... With her direct offer to the Kin, she's pulled yet another "Seafolk Bargain" that Egwene will be forced to cover up, unless she decides Elayne is due for a lesson and lets the Hall deal with that.... Elayne sure could use an experienced Gray as Aes Sedai watchdog, she struggled over and over to draw the line between her friendship to Egwene, her responsabilities to the Tower as Aes Sedai, and her legal powers as Queen of Andor.

Mind you, this doesn't change the fact Elayne and Nynaeve often have better ideas or visions of what Aes Sedai should be, but in matter of authority, the image of the Tower and diplomacy, the dynamic duo is trouble, and unlike what Egwene thinks, Elayne far more so than Nynaeve ever was. She's been the bad influence on Nynaeve, not the other way around.