There are such plants. And in the region where I am from they where traditionally grown before growing wheat or barley. They where not edible but they bring a lot of nitrogen into the ground. Growing nothing edible for regenerationin in my opinion is a clear sign that nitrogen was the limiting factor for agriculture.
There are also such plants which do produce food for humans: quite a lot of them. All of the beans and peas, just to start with.
Nitrogen can be a limiting factor; but it's not nearly as limiting as the fertilizer sellers want everyone to think.
OK, that I can believe. But the minor problem with this scenario is that that entire vault in Svalbard isn't going to simply hand you proven-suitable crops for an ecosystem that just fell out of the mixer and has never existed on the planet before. In order to solve a problem of that magnitude, we'd need experiments, cross-breeding, possibly downright genetic manipulation, possibilities which the post-Rash society, reduced to subsistence farming, may simply not have ...
The same problem's going to exist without Svalbard, though. Increasing the stock people have to work with, by adding in seed from Svalbard, wouldn't make it easy, but it would make it more likely to succeed.
I presume the seed stored in Svalbard includes descriptions of what's known about each variety; I doubt people would have to take things at random with no idea whether they were likely to be suited to Egypt or Iceland. They'd choose samples of varieties known to produce in difficult conditions, and in the type of difficult conditions they were experiencing. True, a lot of them wouldn't work well -- but that would also be true of what they'd scavenged out of gardens and very true of what they'd scavenged out of farms.
And experiments and cross-breeding are what farmers have been doing ever since the first people started farming. Subsistence farmers do that all the time. That's how they got all those varieties of potatoes bred for best production in each microclimate on each soil type; plus which, of course, that's how we got just about all the crops we now rely on for food, in anything remotely resembling modern form.
ETA: there are also varieties that do better than others at dealing with variable conditions. Many modern agricultual varieties are what I call "perfect conditions" crops: if they get just the right recommended dose of NPK, the right amount of irrigation water, the right schedule of application of insecticides and fungicides and herbicides: lots and lots of crop! (Very likely tasteless and low nutrition per bite, but lots and lots of crop.)
But plant those things in erratic conditions and/or when those inputs are unavailable, and they're worthless or next to worthless; while a variety that was bred many years ago to keep people from starving even when things weren't ideal won't ever produce quite as much as the perfect-conditions crops will if they get those conditions -- but will produce a fair amount of crop under a wide range of conditions.