Is there any diachronic reason whereby synthetic perfective passive forms like *amavitur (and similar ones) are not possible and analytic forms like amatus est (and similar ones) are selected instead? Since passive and perfect can each appear on their own (e.g., cf. amatur and amavit, respectively), why can't they be combined and give a form like amavitur and/or why should an analytic form like amatus est be used instead?
In Proto-Indo-European, there were multiple complete sets of person-number markings, used for different tenses of the same verb. You can see the relics of this most clearly in Ancient Greek, where the present tense conjugates -ō -eis -ei, the aorist tense conjugates -a -as -en, and the imperfect tense conjugates -on -es -en.
In Latin, one set of endings was used for the present tense, and a different set for the perfect. The cognates for these are the Greek present and aorist endings, respectively, though evolution has made them hard to recognize: the cognate to Greek's first-person singular aorist -a has been augmented with a particle -i and the -a-i then contracted to -ī, for example, to give Latin amavī.
Ancient Greek has mediopassive equivalents to all the endings I listed, and it seems that PIE did too. But in Latin (and actually Italic in general), for whatever reason, the passive equivalents to the perfect endings didn't survive. Since those endings were fundamentally different from the present ones, though, it didn't make sense just to apply the present's passive forms. So they made do with the periphrasis you mention. We don't see *amāvitur for the same reason we don't see *amistī for "you love": even though the two sets of endings have some similarities, they're fundamentally different, and weren't seen as interchangeable.