I wonder if it's a Germanic remnant.
English literature is predominantly past tense, while German literature is largely present tense.
English is basically old German and French mashed together. When the Normans invaded Britain, the Germanic residents became their servants and slaves. Now, the languages didn't merge equally. There was a socioeconomic divide. For example, the Normans didn't care for their own animals, they made their Germanic servants do it. Hence, most English words for animals come from German. Likewise, fine food was the privilege of the ruling Normans, and thereby many English words for foods, especially complex dishes, come from French.
Jokes are primarily oral traditions, and they're also often considered "boorish" or "uncivilised." So, it seems plausible that in the early days of English, more jokes were told by the Germanic servants than their affluent masters, and were more often spoken than written down. So while written English adopted the past tense, spoken jokes retained the Germanic present tense.
This is just conjecture. But it seems conceivable.