I'm sure lots of fans of drop tunings and newish or older metal with drop tunings may disagree, but I think you're right to focus on a default EADG rather than experimenting with other tunings. Although the different tunings probably would help you learn the full range of the instrument, personally I think it's a good idea to learn that basic tuning inside out before branching out. It is a bummer it keeps you from playing some of those songs, but there are a gazillion heavy songs in standard tuning that can keep you busy in the meantime.
I also know people who swear you have to have a 5-string bass.
Different strokes and all, but i feel like I have a lifetime of learning to do with my measly little 4 strings and boring old standard tuning.
Until I get my BEAD bass finished, I detuned my two players to DGCF (a whole step down). Patterns are the same, just two frets up for songs in standard tuning. Works great with EBCFs on my Thunderbirds. May or may not work well with your strings.

