The r in the endings of verbs comes from the latin, very often : aimer/amare, laisser/laxare, passer/passare. The exact cause of this remaining r that is not pronounced, which I wouldn't consider as a trend towards non rhoticity but as a local phenomenon of non rhoticity is to be found in the evolution of the pronunciation independently of a rule of liaison. This exact cause I, personally, do not know. It is both a local phenomenon of non rhoticity and just another instance of the usual French liaison, although there is nothing very usual about liaisons: sometimes they should be effected, sometimes they are not allowed. The liaison is effected because the r is found at the end of the word ; I suppose the principle that brought that practice about is a little bit as what you call hypercorrection in English phonetics : numerous cases exist where two vowel sounds are juxtaposed and no intrusive r is introduced in French, to the contrary of the modern English trend ("vanilla ice" pronounced as "vanilla rice"); for instance you effect that liaison normally in "la première arrivée" because the r is pronounced; you also do in "le premier arpent" although the r is not pronounced in "premier".
- été évanouie
- avait éternué (the t is not pronounced)
- passé à peine révolu
We can't talk of a trend towards non rhoticity because r is consistently pronounced before consonants (marteau, partie, carte, perte, Orly, pèrle…) and in the endings (mère, père, terre, arrière, faire, finir…). The r's not pronounced in the endings occur in great number in the case of the ending "er", specially in the verbs in the first group, and in many nouns (fumier, poirier, amandier, roncier, foncier, atelier…).
Interesting, no liaisons (at least for those): fumier étendu, poirier élagué, amandier atrophié, roncier épais, atelier en ville,
- In fact, in singing this r in the verbs from the first group (ending in "er") is sometimes pronounced without there being any need for a liaison, that is before a consonnant. This is true for the songs of a more traditional sort and in the past century; it's probably not found in modern singing of a few decades past. There does not exist an accent in which this r should be pronounced consistently; the pronunciation of ending "er" is uniformally the same as that of é in the whole of France.