I don't think Jesus and Paul are making the same point.
Jesus is talking about being distinctive by contrast to the rest of the world. His analogy with salt comes in the midst of two other analogies that bring his point into sharp focus as we consider what the saltiness of food has to do with a city set on a hill or a light put under a basket.
You are the salt of the earth; but if the salt has become tasteless, how can it be made salty again? It is no longer good for anything, except to be thrown out and trampled under foot by men. You are the light of the world. A city set on a hill cannot be hidden; nor does anyone light a lamp and put it under a basket, but on the lampstand, and it gives light to all who are in the house. Let your light shine before men in such a way that they may see your good works, and glorify your Father who is in heaven.
The thing about salty food is the fact that you can't miss it. If the food has salt in it, we taste the salt. And salty food has a distinct flavor that is hard to miss.
How does salt become tasteless? In NT times, the salt wasn't pure salt like we have today; It was a mixture of salt and other minerals. And when the salt mixture got wet, the salt leached out leaving only the impurities behind. If this happened to the salt, the salt would lose it's taste and wouldn't be good for anything else and so it was thrown out on the ground.
Jesus is saying, you might be like the salt mixture that still has salt in it; or you might be like the salt mixture that has no actual salt in it. Just as the salt mixture containing salt is distinctive from the mixture without any salt, you are to be distinctive in some way.
The next analogy Jesus gives involves a city set on a hill. In this analogy, the city set on a hill can not be missed. It can be seen from miles away, even from far off in the distance. And if we light up the city, especially at night, the city on the hill can be seen at night in the dark.
The final analogy incorporates the lamp set in a candlestick, which is a light placed in an elevated position out in the open. Just as a city set on a hill is elevated, exposed and easy to find, a lamp set in a candlestick is elevated, out in the open, and shines light on all its surroundings. We don't put lamps under a bushel because this would defeat the purpose of the lamp. We want the lamp to be up high and out in the open so that it will shine its light on everything in the entire room.
And so Jesus invites us to ask, what does a salty salt mixture have in common with a city set on a hill, and a lamp set in a candlestick? The common theme that ties them all together is the difference between what is concealed and what is revealed. Each of these items is conspicuous in some way. Salty salt is easily tasted; a city set on a hill is easily noticed; and a lamp set in a candlestick makes everything else observable.
By analogy then, Jesus challenges us to be beneficial through our willingness to be both vulnerable and uncommon. Just as salt is good when it continues to produce an unmistakable taste, we are to be good in some way analogous to that. Just as a city set on a hill can't be hidden, we ought to be those whom others recognize by taking the higher ground; and just as a lamp set in a candlestick, we are to enlighten those around us so as to be clear, understandable, factual, and true, standing in contrast to our culture while we critique it.
The Lord is our prime example, who both made himself vulnerable, and remained uncommon as he spoke to his people and taught them another, higher way. He didn't put his light under a bushel, he stood in contrast to his culture and made his ideas known. He set himself as a city on a hill, which was as overt and obvious as possible. And he remained as distinctive and useful as salty salt, which never lost it's taste.
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