I'm curious what you mean by not looking "period". The HPLHS fonts frankly seem to just be poor quality, rather than old. If you look at the images of the original type, Doves appears to be quite faithful to the original. Perhaps it's worth noting that we still use typefaces remarkably similar to the Romans, particularly Times New Roman, which despite its many shortcomings retains a "modern" look by virtue of still being in use.
Roman type has roots in Italian printing of the late 15th and early 16th centuries, but Times New Roman's design has no connection to Rome or to the Romans.
It was created by the descendants of the Romans, in the same physical location as Ancient Rome, and based on the numerous examples of letters that were still around on Roman buildings.
If that is “no connection” what exactly would a “connection” look like?
I'll admit I'm no typeface expert, but this seems to miss the point. Wikipedia's own page on Roman type [0] says "Roman type was modelled from a European scribal manuscript style of the 15th century, based on the pairing of inscriptional capitals used in ancient Rome with Carolingian minuscules". And visually, there's clearly an influence, though many centuries removed. My point is merely these very old typefaces remain modern looking because we still use similar ones today.
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.
Not all documents from the time period would've had such low quality though, and not everyone would want such quality in a modern document. If you want such an effect, it's always possible to add it later, but it's rather more difficult to remove it if it's baked into the font file.
The H.P. Lovecraft Society has some 19th century fonts, if you need them.[2] Those were recovered from old documents.
[1] https://typespec.co.uk/doves-type/
[2] https://www.hplhs.org/resources.php