In the mid-19th century government mail contracts were enormously valuable (the 1841 contract for carrying mail between Liverpool, Halifax, Quebec and Boston was worth £80,000, which https://www.nationalarchives.gov.uk/currency-converter/ values at around £4.8 million in 2017), and all but essential for companies such as the Royal Mail Steam Packet Company (which became Cunard) operating paddle steamers. The stipulations of mail contracts for a frequent, reliable and fast service made the ships desirable for passengers as well, but passengers alone were insufficient to cover the running costs. As engine and propulsion technology improved liners became more efficient, mail contracts were not so vital, not the primary driver as the size and speed of transatlantic liners became a matter of national pride at the start of the 20th century, but always a useful source of income for companies. The RMS designation of the Lusitania and Olympic was, as you surmise, largely a holdover, mail by no means being the primary purpose of the ships, but they did carry post and post office workers.