I've tried both ways, and had infinitely more success when I spend time to make it more formal. You're right that electronic communication tends to be informal, so that when people send anything by email, they tend to be more informal as well, apparently under the impression that the paradigm has shifted.

However, you make a good point that if you care about a job, you will spend time to work on making a good resume and cover letter.

I don't think it's outdated. It just makes sense. $0.02