Seeing how those same PD-inviolable disasters happened during TOS and TNG more than once per series, I'd say they're happening a lot more than you think and in a greater volume than just a few planets. The Pro-Interventionalists who want to abolish the PD would likely force Starfleet to actively search out the cosmos (within Fed boundaries or not) for every endangered world (no matter what level of danger) and demand they step in. And those ships and personnel wouldn't come from nowhere, they'd come from Starfleet's other duties like defense and such.
And it would sill take hundreds if not thousnads of ships working round the clock for any kind of evacuation/relocation, so even if it was only rare it still wouldn't be economically viable.
I concede that the writers utilize the PD for drama, and probably more than they ought to.
If the Federation economy really would become so strained by the incessant helping out, then a balance would surely be reached. The Federation electorate is a collection of holo-coddled, replicator-fed semi-infants and a smattering of Starfleet superhumans... as long as their quality of life remains unchanged, they'll be willing to help, but if it is affected in the merest, they'll debudget the Starfleet Rescue Division in a heartbeat.
For the same reason, the United States of America is willing to make half-hearted efforts to prevent famine and halt disease. Of course, we wind up doing tremendous good regardless of our half-assedness, but we don't dedicate our every resource to it. Nor, perhaps, should we.
And your reasoning for the protection from natural disasters such as gamma ray bursts is akin to what if random aliens had stopped the meteor from wiping out the dinosaurs. Somehow I doubt humanity would've appreciated that.
Dinosaurs aren't intelligent and intelligent life only existed as a potentiality in the future. The PD doesn't even apply to M-class worlds with no intelligent life on them. If something like Starfleet existed 65mya, they'd have colonized us already with a few hundred libertarian xenophobes in purple jumpsuits.
On the subject of PD violations, the hugest one might have escaped the attention of some compilers: Devil in the Dark. In this, a colony of miners tries to eradicate the strange indigenous intelligence of Janus IV, before being stopped by the Enterprise and Kirk--who suggests that the miners work with the natives, because the miners will benefit from the natives' tunnelling abilities, and apparently because the natives will benefit from not being shot with phasers anymore, but also the advanced technology of the Federation.
And truly, I think Devil captures the real spirit of Trek, which is the seeking out strange new worlds and new civilizations thing. The Horta truly live in a strange new world... with the PD hanging over our shoulder, we never even set foot on Janus IV, or rapidly evacuated once we ascertained the horrendous damage we did to Horta society.
I'd like to raise a final point about the assertion that alien technological assistance might coddle a less-advanced civilization, just because their own capabilities weren't stretched to reach an arbitrary technological level...
No one invents their own technology. Not all of it. This computer I use, the car I drive, the building I live in: these might as well have been designed by an alien intelligence, because they weren't designed by me. Have I been coddled, not to have been given a computer until I could design one myself, just like it is improper under the PD to give a society warp drive until it can build one itself?
To apply a different standard to a civilization than to an individual seems odd. The whole point of a civilization is to raise up its individual constituents who could not possibly command such powers on their own but can do so collectively. The Federation, having realized the truths of equality, recognizes that those who work do so for the good of the whole as well as for the individual.
Sure, there are good reasons to deny access to certain technologies to certain civilizations, yet using self-attained warp as an indicator of civilized potential is rather arbitrary. Look at the Romulans. Or the Cardassians. Or the Gorn. Or virtually any species but the humans and the Vulcans. Or even them, they can suck as badly as anyone.
The warp barrier test simply can't be predicated on the notion that a warp capable society is sufficiently civilized and thus ready to come under Federation tutelage. There is far too much evidence to show that even a civilization which developed warp on its own is perfectly capable of having barbaric ideas about how to behave.
The warp test is predicated, rather, on the fact that once a society is warp capable, the Federation simply can no longer restrain their access to the wider galaxy, and the cynicism that dictates the Federation is better off with bound allies than with potential enemies. Hence we have civilizations which are Federation members or affiliates but who have values contrary to those which the Federation stands for.
The Prime Directive attempts to balance the evils of interference and non-interference but is clearly a failure as a policy. Even when non-interference works, it produces ridiculously bad societies, like the Son'a or Ardanians, as often as it does mature ones, like humans. When it doesn't work, it leads to extinction of unique and valuable life. Maybe we have an instinct to "coddle" because "coddling" works.
I'm curious, would the Prime Directive apply to a society which, through modern-day means ascertained the existence of habitable nearby worlds and sent a generation or sleeper ship to go check it out? Say a big relativistic arkship glides into Sector 001 one day and says "What's up?" Do we pretend we're not home?
I doubt every Romulan loves the smell of napalm in the morning. They might have epic poetry about it, but they don't want to live it. 
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