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Recent literature indicates that attractive interactions between particles ofa dense liquid play a secondary role in determining its bulk mechanicalproperties. Here we show that, in contrast with their apparent unimportance tothe bulk mechanics of dense liquids, attractive interactions can have a majoreffect on macro- and microscopic elastic properties of glassy solids. We studyseveral broadly-applicable dimensionless measures of stability and mechanicaldisorder in simple computer glasses, in which the relative strength ofattractive interactions -- referred to as `glass stickiness' -- can be readilytuned. We show that increasing glass stickiness can result in the decrease ofvarious quantifiers of mechanical disorder, on both macro- and microscopicscales, with a pair of intriguing exceptions to this rule. Interestingly, insome cases strong attractions can lead to a reduction of the number density ofsoft, quasilocalized modes, by up to an order of magnitude, and to asubstantial decrease in their core size, similar to the effects of thermalannealing on elasticity observed in recent works. Contrary to the behavior ofcanonical glass models, we provide compelling evidence indicating that thestabilization mechanism in our sticky-sphere glasses stems predominantly fromthe self-organized depletion of interactions featuring large, \emph{negative}stiffnesses. Finally, we establish a fundamental link between macroscopic andmicroscopic quantifiers of mechanical disorder, which we motivate via scalingarguments. Future research directions are discussed.

Mechanical disorder of sticky-sphere glasses. I. Effect of attractive interactions