What you call "shortcuts" are really trade-offs, and they are inherent in any project, in any discipline, you will undertake. It is inescapable. There is an idiom in the engineering world (yes, I put this in a comment above): You can do it well, you can do it fast, or you can do cheap. In the best case, you get to pick 2, so the decision needs to be made on which 2 are most important for the given circumstance.
To use your situation as an example, you have a small team (cheap) and a tight deadline (fast), so it follows that what you build will probably have a lot of bugs and you may not get to do the testing you want to do. This is the decision your manager has made, and he will have to deal with the consequences. You want to do more testing to build a better product? He will either have to give you more time (lose out on 'fast'), or add more people to do the testing (lose out on 'cheap'). Those just may not be options for the project. This is inescapable.
So quitting your job won't solve anything, because this principle will be there wherever you go. The most you could do is find another employer who maybe puts an emphasis on a different part of the triad, but even that will vary from project to project. They are making decisions just like your current manager.
Any decision made against this triad will have consequences and someone will need to figure out if they are acceptable. Slowing down to build a better product means you might miss a deadline. Adding more people means they won't be working on something else. Building it fast and cheap means spending more time on supporting it later (or living with a shoddy product). Depending on the requirements, any of these decisions might be valid. The key is understanding the trade-offs, what they mean for the business, and making sure the decision-makers understand them. That's one of the most important skills you can have.