Recent observations of debris discs, believed to be made up of remnant planetesimals, brought a number of surprises. Debris disc presence does not correlate with the host star's metallicity, and may anti-correlate with the presence of gas giant planets. These observations contradict both assumptions and predictions of the highly successful Core Accretion model of planet formation. Here we explore predictions of the alternative Tidal Downsizing (TD) scenario of planet formation. In TD, small planets and planetesimal debris is made only when gas fragments, predecessors of giant planets, are tidally disrupted. We show that these disruptions are rare in discs around high metallicity stars but release more debris per disruption than their low [M/H] analogs. This predicts no simple relation between debris disc presence and host star's [M/H], as observed. A detected gas giant planet implies in TD that its predecessor fragment was not disputed, potentially explaining why DDs are less likely to be found around stars with gas giants. Less massive planets should correlate with DD presence, and sub-Saturn planets (\(M_{p} \sim 50 \M_{\oplus}\)) should correlate with DD presence stronger than sub-Neptunes (\(M_{p} \leq 15 \M_{\oplus}\)). These predicted planet-DD correlations will be diluted and weakened in observations by planetary systems' long term evolution and multi-fragment effects neglected here. Finally, although presently difficult to observe, DDs around M dwarf stars should be more prevalent than around Solar type stars.