To get your relic you need to beat Titan Hard Mode which requires an ilvl of 57.

So should you worry about getting the relic for Thornmarch / Moggle Mog? No. Go to your grand company and buy the ilvl 55 weapon. You're going to want to use that to get through the primals anyway.

Should you get your relic before trying the other post-50 dungeons that require an ilvl of 60+? Yes. You can run Garuda, Ifrit, Thornmarch, and Titan to get ilvl 60, 70, 75, or 80 weapons but the relic is the best way to get an ilvl 90+ weapon.

Once you've gotten your relic up to 90 it's up to you if you want to continue it. Right now it is easier to buy the ilvl 100 soldiery weapon and upgrade it to 110 than it is to get your relic up to 110. Buying the soldiery weapon is 1300 soldiery and a drop from Syrcus Tower/Second Coil. Getting the relic from 90 to 100 needs 9 books at 500 soldiery each=4500 soldiery. But currently getting your relic from 110 to 125 is arguably easier than buying the ilvl 120 poetics weapon and upgrading it to 125 since poetics has a weekly cap and you need a drop from Final Coil. By the time the expansion lands this summer the poetics weapon will probably be as "easy" to get as the soldiery weapon is now.