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There seemed to be no reason why everyone couldn’t just use the same prime, and, in fact, many applications tend to use standardized or hard-coded primes.

Then, if the prime number is standardized or hard-coded, why they just not use it? Why we need to break it?

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It shouldn't be nor standardized nor hard-coded because someone with the funds (e.g. NSA) would need to break the encryption using this 'standardized' number only once.

If everyone used a random very large prime (spec suggests so) then NSA would have to break with every prime number possible which currently is not possible

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