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 a stereotypical accent today often did originally have the speakers, but its real-world use gradually faded away with the lapse of time only to remain in people's memory.
So-called Standard Japanese was established in the 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 thought people should talk like.
What you mentioned was 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 the Edo period.
After the Meiji Restoration, Kansai nobles and VIPs migrated from Kyoto to Tokyo in crowds, giving people the impression that their way of talking was how typical respectable men talk. EDIT: After rechecking 金水 (2003), turns out that the educated upper class in Edo was speaking Kansai dialect while commoners used native Edo dialect throughout the 19th century, which led to the perpetuation of this stereotype after Meiji. (Thanks to @user4092 for correcting.)
A bit off on a tangent: today, words such as おい and こら are used to scold somebody in Standard Japanese. They are actually "you" and "hey" in Satsuma dialect, respectively. The majority of police officers in Meiji Japan were from Satsuma Domain, as they had been one of the main factions that overthrew the Shogunate. As a result, their accent had been recognized as oppressive talk characteristic of a police officer.
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 person in Japan uses two or more first-person pronouns depending on the situation (technically they're not even pronouns in Japanese, so, I guess two or more "ways to address oneself"?). I daren't say the shift to わし is common, but people can easily change their normal "pronoun" as a result of influence from those around them.
Totally unrelated but I have an acquaintance who uses わし in casual settings (in Tokyo dialect) for some unknown reason.