The only time you should worry your users about something that happened in the code is if there is something they can or need to do to avoid the issue. If they can change data on a form, push a button or change a application setting in order to avoid the issue then let them know. But warnings or errors that the user has no ability to avoid just makes them lose confidence in your product.
Exceptions and Logs are for you, the developer, not your end user. Understanding the right thing to do when you catch each exception is far better than just applying some golden rule or rely on an application-wide safety net.
Mindless coding is the ONLY kind of wrong coding. The fact that you feel there is something better that can be done in those situations shows that you are invested in good coding, but avoid trying to stamp some generic rule in these situations and understand the reason for something to throw in the first place and what you can do to recover from it.