Olivia's only use is Rally Spectrum? Hardly. There are quite a few maps in this game that are fairly small, even as far as into the lategame (Yen'fay's chapter, for example), and frequently your units start right next to the enemy. Take Fort Steiger for example. There are four enemies on the left path, right next to your units, 5 on the right path, and 2 down the middle. You have, if I remember correctly, 4-5 units that you can place to the sides, and 4 in the middle. That probably won't be enough to kill the ones on the flanking paths, especially since you might need to pair some units up to double/avoid getting doubled (the Valks have like 22-23 Speed even on Hard Mode), but killing them on the first turn is quite advantageous because then you can put your frailer units much further ahead right away. So you use Olivia and have the unit with the highest offense go twice in a row. Now you can finish the enemies on the first turn (for whichever side she's on), and more than likely that side will be cleared up. Situations like these are fairly common, and using Olivia can give you a boost in turn count; what other unit, besides Avatar and earlygame Fred, can claim to singlehandedly boost turn-count as well as Olivia? Especially since Hard Mode is easier than Lunatic, and unit differences are much smaller. There's not a lot of difference between fielding Sumia and fielding Cordelia, for instance, whereas there is a lot of difference between fielding Olivia and fielding one of those two.
Which brings me to another point. Olivia isn't just replicating the function of another unit; she can replicate the function of a paired unit too. Your Dark Knight Miriel is just not fast enough to double, and you have a nice +8 Cordelia pair waiting in the wings, but can't use it because you still need Cordelia's attack? Easy. Pair Cordelia with Miriel, have Miriel attack, and have Olivia refresh Miriel. Now you get two souped up attacks instead of two much weaker and less useful ones, and one of your units didn't have to attack and eat a counter (Cordelia, in this case).
What about a unit with Galeforce? Kill, kill with Galeforce, get danced for, kill, kill with Galeforce again. Now Olivia isn't just adding +1 action; she's adding +2. Or how about rescue chains? You can pair up Panne and Fred, attack, then refresh Fred so he can drop Panne off with Avatar, and attack another powerful enemy. Or healers. Olivia can allow Libra to attack, then heal another unit, or heal another unit twice. Or Olivia can allow a frail unit like Lissa to attack, eat a counter that leaves her at 5 HP, then quickly pair up with Fred or someone for safety. The range of possibilities is quite vast. No other unit offers you that kind of flexibility.
Even if that weren't the case, fielding a 13th or 14th combat unit is far less useful than fielding Olivia. You often won't even have enough space for that combat unit to be attacking an enemy, so Olivia refreshing a unit is automatically more valuable in that case. And, assuming that your top spots are filled with the best combat units, that combat unit will be significantly worse at attacking, anyway, so it's likely better off just being paired with someone--and guess what, in that case, Olivia can just refresh a stronger pair. Done.
Let's also not forget that, besides durability, Olivia is the least reliant on stats to do her function. She's just as efficient at dancing at level 1 as she is at level 15, so you can field her on a need-to-use basis, rather than consistently (though I don't know why you wouldn't use her consistently, considering that dancing is just that good). Even with that, though, she's drawing from her own separate pool of EXP that no other unit has access to, much like healers (except that healers still need to burn staves that other healers could be using in order to get that EXP, so even then there's a little bit of a drain), so you're still better off fielding her than a 13th combat unit who will take kills from others, etc.
And in the process making said competition considerably worse?
Are we ignoring opportunity cost? Just because Panne is the best use of a resource doesn't mean that there isn't a negative associated with giving her that resource. +10 -5 is still only +5, not the full +10 that using the resource without a cost would give.