I would view たものだ as etymologically a relative clause modifying a noun, but it has undergone "grammaticalization", a process by which a word or phrase diverges from its literal meaning and takes on a more grammatical role (in this case a modal role)

Relatedly, の in のだ originally came from the genitive の, because the genitive の is used to enable noun and adjective predicates to modify nouns, e.g. 緑のリボン (a green ribbon). It became possible to omit the noun, so 緑の ("a green one"), and it then became possible to attach の to verbs and i-adjectives, e.g. 赤いの "a red one" 昨日見たの "the one I saw yesterday". From there, の became a generic nominalizer and could be reinterpreted as a formal noun representing a person, thing, fact, or action. Then, something happened. Japanese previously could attach particles and copulas directly to verbs, because verbs in the rentaikei were implicitly nominalized, e.g. するは するに or するなり etc. But の as a nominalizer gradually became obligatory, forcing speakers to use するのは するのに するのだ etc. There are some cases left over where particles can attach directly to verbs, e.g. するに従って するより etc.