but there was not really any concept of store and forward like you have with texts and email.
Notice how they had to tell the operator to make sure there was a clean line, then manually sync the two devices.
I guess you could claim the telegraph network had something akin to store and forward, as each operator would note the message, and then relay it down the chain.
Howard Krum finally figured out how to do sync right in 1919, when he invented serial stop-start async as we know it today. In 1924, he came out with the Teletype Model 14 tape printer, with the first reliable mechanical async decoder. That design was used into the 1960s, and most computer serial ports are compatible with it. (45.45 baud, 5 bits, no parity, 1.5 stop bits.)
Telegraphy was store and forward, but manual at first. By the late 1920s, punched tape was used, but someone had to manually look at tape, tear off each message, and put it in the proper outgoing reader. This was manual Sendmail. After WWII, automated forwarding centers were built, with paper tape punches connected to readers via a bin as the storage mechanism and telephone-like switches to change connections. These were hugely expensive to operate for the amount of data transferred, but it was essentially email, with electromechanical technology.
(I restore Teletypes as a hobby. Here's a Model 14 working.[1]. If you want to see some in action, they'll be at the Clockwork Alchemy Steampunk Convention, Memorial Day weekend, in San Jose.)
(Edit: in fairness, that would be more like "land line texting" than email.)
That said, it seems most of them were operated over dedicated networks. And the one i can find a reference to that was operated by AT&T had their own area codes etc.
All in all i get the impression that it could be done, but likely only so at great expense. Consider that having a ticker tape printer in your home suggested you were someone with great wealth that was watching the stock market 24/7.
Even having a telephone was not chump change. My parents talk about having one phone for several households, with the bell patterns used to indicate who it was for.
I suspect we should not underestimate how much the IC and packet switching has driven down cost of installation and operation.