Since when were summons weak?
In Final Fantasy III, there were two types of summoning magic: the magic used by the Conjurer class, and the magic used by the Summoner class. The magic used by the Conjurer class could be unreliable, since the summon would use one of two different abilities at random. However, some of those summons were still pretty useful. For example, Titan's two options were "powerful physical attack" and "another powerful physical attack". Ifrit's was "do good fire damage to all enemies" or "really strong heal on the the whole party". The Summoner class used the summoning attacks that are more familiar from the rest of the series: Diamond Dust, Judgement Bolt, etc. However, the sheer quantity of spell uses the Summoner gets for pretty solid attacks makes it a pretty good class.
Now, Rydia's summoning magic was also pretty good. If you compare Bahamut and Meteor, for example, Bahamut has 60% of the Mp cost and one third the cast time to get the same rough effect: do max damage to all enemies. It is a phenomenal spell in FFIV, and Summon magic as a whole is along the same lines.
Now, Final Fantasy V is where the Summoner really shines. Summoner is a wonderful class in FFV because of two things: the ability to trivialize every random battle in the game, and excellent defensive utility. The FFV Summoner gets Golem, a summon that negates all enemy physical attacks, which is just as good in practice as it sounds, and Carbuncle, which gives the whole party reflect. The FFV Summoner also has Phoenix, which is probably one of the best single spells in the game.
Summon magic in these first three games (as well as in most later FF games) has one very useful feature that differentiates it from Black Magic: Spell Attack power isn't cut in half when targeting more than one enemy. This makes Summon Magic for the most part strictly better than normal magic for fighting groups of enemies. And Summon Magic gets the job done. Between Ramuh, Titan, and Syldra, you can pretty much wipe out any non-boss enemy group in FFV with just one spell. That is the biggest selling point of Summon Magic.
Now, they significantly nerfed Summon magic in FFVI, thanks to the imposition of the "once per battle" limit. That doesn't stop summons from being enormously useful, thanks to added variety and support abilities.
To be honest, Summon magic began to go downhill in FFVII. While the summon spells themselves are just as great as always, the shift begins towards spectacle over substance. And that is the big curse that has been plaguing summon magic since: they make the spectacle bigger and bigger, but at the cost of making summon spells less and less purely functional abilities.
This hits its pinnacle in FF XII where summons are completely over the top in spectacle, thanks to music shifts, large size of the summoned monsters, and flashy attacks. The FFXII summons are way, way overcosted (your entire MP gauge and two party members) for something that doesn't actually do much. They are completely useless. In fact, they are actively detrimental to you. They are only useful for very limited and atypical situations. Summons in FFXIII continued that same trend: flashy, expensive, and mostly useless. I think I'll like the whole "summons fighting beside you" idea once they actually make it fun and well balanced.
Final Fantasy XIV's approach is somewhat underwhelming as it is, thanks to the weak focus of the class on its pet. From what I gather, FFXI was a lot better in this regard, though at the cost of not giving the player much to do. On the plus side, if you just want summons for spectacle, FFXIV has that in spades thanks to the Primal battles. Some of the new Primals and redesigns of classic summons are excellent too.
