In English, we know how to talk like a pirate, even those classical pirates are no more around (aye, pirates still exist, but they tend to speak Somali or whatever...). What is stereotypical accent today often did originally have the speakers, but gradually faded away with the lapse of time only to remain in people's memory.
So-called Standard Japanese was established in Meiji era based on Tokyo dialect, which is a well-known fact. That means, the word usage more or less reflects what Tokyoites at that time think people would talk like.
What you mentioned the stereotypical "old (wise) man" talk such as 「わしは～～じゃ」 is, in fact, a living dialect in some Western Japan areas. As Kyoto has been Japan's traditional capital, the most prestigious dialect was that of Kansai until the end of Edo period.
After Meiji restoration, those nobles and VIPs migrated from Kyoto to Tokyo in crowds, struck people that their way of talking was how typical respectable men talk. EDIT: After rechecking 金水 (2003), it proves to be that the educated upper class in Edo was speaking in Kansai dialect while commoners used native Edo dialect throughout the 19th century, which led to fixation of this stereotype after Meiji. (Thanks to @user4092 for correcting.)
A bit off on tangent, but there are words おい and こら we typically use to scold somebody in Standard Japanese. They are actually Satsuma dialect of "you" and "hey", respectively. The majority of police officers in Meiji Japan were from Satsuma domain, as they were one of main factions that overthrew the Shogunate. As a result, their accent had been recognized as oppressive talk a police officer would do.
People just decide at some point in their life to start using a pronoun that matches their old age. Seems unlikely.
It's not that unlikely, as almost every people in Japan use two or more pronouns according to the situation (technically they're not even pronouns in Japanese, so, two or more "ways to address oneself"?). I daren't say shift to わし is something common, but people can easily change their normal "pronoun" following those who around them.
Totally unrelated but I have an acquaintance who uses わし in casual settings (in Tokyo dialect) for unknown reason.