EDIT - I just looked at the wiki and it gives an answer that surprised me. Can't find any source so this may be an interpretation more than anything else, but if anyone can find any canon reference that'd be great
ANSWER - It's REALLY hard.
According to the wiki, the High Summoners we hear of in the game are the ONLY ones ever to accomplish the task of defeating Sin, rather than simply the ones we hear of. As such it suddenly becomes much more obvious why no-one could step up to help Braska - Spira only produces a High Summoner (on average) every 250 years. As an interesting side-note, the wiki cites Final Fantasy X Ultimania Omega when stating that Braska's Calm only lasted a year (which makes sense, considering the attack on Besaid that happens at some point before the game begins).
Original answer follows...
As far as I'm aware this is never directly addressed in-game, so might have to stick with speculation here.
When speaking of Yuna, Wakka says she had "the talent" and so trained to be a summoner. The implication here seems to be that it's not just a matter of training - one must have an innate gift and the will to learn the skills. Depending on how rare this ability is, we immediately have a reason why there are so few summoners. And if we consider that line of thinking further...
Number of Summoners
While it is true we several summoners on the road at the same time as Yuna, it is worth noticing that they are relatively few in number. Of those that have the gift, not many would choose to be a summoner, for obvious reasons. Beyond this, we also hear of the dangers involved - more than once you stumble across the last resting place of a would-be summoner, or meet someone who tried to become one and gave up, either due to the danger or the sheer strain on their will/abilities. So consider that the group of candidates has been whittled down even further, and now look at...
The pilgrimage to Zanarkand is not simply a tour for the fun of it - it builds the summoners skills and power so that they are ready to get the Final Aeon. It is possible that, during the Calm, the power and skill fades to the point that the pilgrimage must be taken again (since no one would want to get the Final Aeon during the Calm, again for obvious reasons).
So, in short, we have an extremely small number of people at any one time attempting a difficult task, one that they may not be able to prepare for ahead of time. This might explain why another summoner wasn't ready to step up to the plate.