The capital letters were indeed inspired by Roman monumental inscriptions. But all the lower case forms were taken from Carolingian designs. Humanists wanted to copy Roman forms to go back to what they saw as writing uncontaminated with medieval influence, but the texts of Roman authors they used to do so were not actually written by Romans but copied by Carolingian-era scribes. It's why its generally much easier for us to read ninth-century texts than, say, earlier (e.g., Merovingian chancery script, yikes) and later scribal hands (e.g., late medieval Gothic).

"If my grandmother had wheels, she could have been a bicycle." Serif type is based on the use of chisels to carve rock. ANY other semblance is purely speculative. The trademark for Times New Roman is owned by the British, not the Italians.

Look up the history of how Stanley Morison and Victor Lardent created Times New Roman for _The Times_.

The connection you are looking for is covered by Fra. Edward Catich in his books, and carried forward digitally in Carol Twombly's Trajan.