That’s because hardly is hardly an adverb. It’s instead a negative, just like not, and negatives go before the verb in modern English.
But that’s just part of it. Being a negative, hardly therefore follows the same special and complex syntactic rules attached to all negatives. So do scarcely, barely, little.
That’s why the first of these is ungrammatical in standard English:
- He has ❌not hardly started yet.
- He has ✅not started yet.
- He has ✅hardly started yet.
The actual adverbial use the OED today considers rare and archaic. Their only citation for using it as an adverb of manner in the past century is this one:
a1973 J. R. R. Tolkien Silmarillion (1977) 273
Isildur came at last hardly back to Rómenna and delivered the fruit to the hands of Amandil.
About its negative use, they redirect you to a note at barely, which reads:
The adverb qualifies verbs, adjectives, adverbs, and adverbial phrases, and esp. numerals (nouns, adjectives, and adverbs), designations of quantity, and indefinite pronouns. In many cases it may most correctly be regarded as qualifying the whole predication, though placed in proximity to the word in the sentence to which the qualification chiefly relates.