Tesla has been reaching out to its earliest customers -- Roadster owners -- in recent weeks, acknowledging that service for their vehicles hasn't been ideal, and promising them a new "dedicated channel" with its own service advisors, technicians and repair centers. In an e-mail shared with CNBC by multiple owners of the original Roadster, Tesla's President of Automotive, Jerome Guillen, wrote: "We appreciate your continued and pioneering support for Tesla. We realize that we need to improve service for Roadster." Guillen also invited Roadster owners to trade in their old cars and put the value towards buying a newer-model Tesla, or a reservation for a next-generation Roadster. As Tesla scales, service can become a bigger potential revenue and profit center for its business. But it's not clear whether service will become a selling point for the company any time soon. During Tesla's 2019 second-quarter earnings call, Musk acknowledged the daunting task before him. He said: "Obviously if we're doubling our fleets every year, managing service, it is quite difficult." Tesla did not immediately respond to requests for comment.

Roadster owners felt left behind

As CNBC has previously reported, many of the electric car enthusiasts who purchased Tesla's debut vehicle felt neglected after the company began to focus on the Model S in 2012, and other models after that. The original Tesla Roadster was made and delivered to customers from 2008 to 2012, meaning the cars are between seven and 11 years old today. Because the average age of vehicles in use in the U.S. is around 12 years, most automakers work to ensure their cars can be easily repaired for a dozen years or more.

But, as Tesla employees and former employees told CNBC, Tesla no longer manufactures spare parts for its original Roadsters, nor does it enable drivers to book Roadster service appointments through the same apps that other Tesla customers can rely on. Delivering great service is increasingly important for automakers as new vehicle sales growth is slowing down overall, says Hans-Werner Kaas, Senior Partner at McKinsey and Company. "OEMs [automakers] are taking a holistic view of service that includes the parts they offer, distribution channels, and in some cases non-dealer repair shops or service points," Kass wrote in an e-mail to CNBC. "High-end OEMs constantly think about the customer service experience and view it as a key driver of customer loyalty. They seek repeat purchases rather than the sale of more parts."

Jerome Guillen, vice president and head of sales and service for Tesla. Uli Deck | AP