I think the part about "caring about only your part of the work" is partially wrong, in that every team member, not just management, should care about every component they are aware of, not just what they build. Everyone should choose to depend on better quality components, and discuss with teammates what those are. If a tool has a defect, tell your teammates about avoidance tactics or workarounds if you can't fix it outright. Management needs to be informed, and if management can't delegate completely (e.g., if money past a threshold must be spent) then a person on the team should develop the cost/benefit alternatives.
So I wrote this back in the day when I was leading a small startup, I think the advice is still valid and can help others. Cheers! :)