"Slave economy" is a misnomer: slaves were numerous (though still a minority) in Italy, but far scarcer elsewhere: Scheidel estimates the total at around a tenth, meaning that most work had to be done by non-slave labour. So it's probably more accurate to characterise both the Empire and medieval Europe as predominantly peasant economies since on this showing the free rural population was 75-80% of the total.

So the real hurdle is the transformation of nominally "free" cultivators into bound serfs. This began already in 4th-century Rome with the stipulation that an agricultural tenant should after thirty years be bound to his plot in perpetuity as a colonus, a practice which carried over into the Byzantine east.

The institution's evolution in the "barbarian" west isn't clear. Ties of loyalty and material support clearly existed between chiefs and tillers, and must have done so previously: Bloch cites the occurrence of personal names in place-names and the survival of old Roman terms in later usage as evidence of local lordship and some degree of continuity. But details seem scant until we reach the 9th century, by which time the classic knightly feudal order is already taking shape.

So much of it's probably about the newcomers adapting some of their own arrangements to local institutions. Slavery persisted on a reduced scale, with slaves often being given their own plots to till for the lord as servi casati ("hutted slaves"): the status only expired in the 10th-12th centuries (largely through Church opposition), with England slow to part with it.

The "slave mode of production" thus never needed undoing, it hadn't existed in the first place. The west's new rulers presumably rejected it for much the same reasons as their Roman predecessors, namely that it required stricter control than a semi-free peasantry, for little if any gain in productivity, possibly the contrary. Europe remained a land of peasants.