  Recent literature indicates that attractive interactions between particles of
a dense liquid play a secondary role in determining its bulk mechanical
properties. Here we show that, in contrast with their apparent unimportance to
the bulk mechanics of dense liquids, attractive interactions can have a major
effect on macro- and microscopic elastic properties of glassy solids. We study
several broadly-applicable dimensionless measures of stability and mechanical
disorder in simple computer glasses, in which the relative strength of
attractive interactions -- referred to as `glass stickiness' -- can be readily
tuned. We show that increasing glass stickiness can result in the decrease of
various quantifiers of mechanical disorder, on both macro- and microscopic
scales, with a pair of intriguing exceptions to this rule. Interestingly, in
some cases strong attractions can lead to a reduction of the number density of
soft, quasilocalized modes, by up to an order of magnitude, and to a
substantial decrease in their core size, similar to the effects of thermal
annealing on elasticity observed in recent works. Contrary to the behavior of
canonical glass models, we provide compelling evidence indicating that the
stabilization mechanism in our sticky-sphere glasses stems predominantly from
the self-organized depletion of interactions featuring large, \emph{negative}
stiffnesses. Finally, we establish a fundamental link between macroscopic and
microscopic quantifiers of mechanical disorder, which we motivate via scaling
arguments. Future research directions are discussed.
