For about 100 years or so, stock quotes (symbol, price, volume) were transmitted by telegraph. Many traders of that era were tape readers, often sitting in the office of their broker before phones were in wide usage. This was replaced by electronic communication networks (ECNs) circa 1980.
Until about 10 years or so ago, 'open outcry' was the method for communicating orders in trading pits. It involved hand signals and/or verbal statements. Floor runners were often used in the transfer of buy or sell orders (by phone to the floor of the exchange) from the brokerage firm to the floor traders. A classic example of open outcry is the trading of orange juice futures in the Eddie Murphy movie "Trading Places." Good cast and a movie worth watching.
For stock trading, in the days before the internet, people used telephones to trade. This was problematic in several ways. First, you had to wait for a broker and on busy market days, you had to wait, wait, wait. An extreme example was the crash of 1987 when the entire system overloaded. You were lucky to even get through to your broker that day. As an aside, I had half a dozen or so covered calls written on Bear Stearns that expired in-the-money on Friday, the last trading day before the crash. It took my full service broker Paine Webber over a week to determine if I had been assigned as expected or not.
With the advent of touch tone phones, retail traders could use it with brokers who offered automated quotes systems. For example, on your phone's keypad, the number 2 has the letters A, B, and C. If I wanted to get a quote for a stock with the symbol ABC I would press 2-1 (A), 2-2 (B), 2-3 (C) with the 2nd number of each pair representing letter's position on that key. Tedious, to say the least.
To the specifics of your question. People yell all the time, for many reasons, in person or on the phone. It's not a farfetched movie scene for a person losing money in the market or fervently desiring to make a stock purchase to be yelling at his broker over the phone.
Pre-internet, if you had a brokerage account, everyone had a broker, either personal or when discount brokers arose, a random stockbroker at your brokerage firm. You didn't have to be rich.
FWIW, rich people tend to utilize money managers who either directly trade the account for them or offers investment ideas to the account holder. However, nothing stops them from calling their broker to effect trades.
You young-uns have no idea what a pleasure it is to trade in the information age :->)