Gilga's Eldritch Knight is STR-based; in fact it dump-stats INT even though it's wizard-based and uses Divine Might instead of Know the Angles. However his build predates the changes to Divine Might (i.e., back when it provided an Insightful STR bonus), so now there's no functional advantage to DM over KtA (other than being cheaper AP-wise). So you're not the first to hit on the idea of ditching DCs on an EK build.
I count five acid spells with no DCs: four single-target and one AoE. But there are still several acid spells with DCs, including two of your Earth SLAs and epic abilities like Energy Blast or Hellball. Plus a couple other useful Conjuration spells with DCs like Web, Glitterdust, and Trap the Soul. So I don't know that I would say Earth Savant "doesn't care about DCs," only that that they have more spells which aren't DC-dependent than other Savants.
The other main issue would be lack of feats: wizard provides up to five bonus feats, sorcerer provides zero. Difficult to squeeze in full melee DPS and caster feats.
That said, you could test your idea with e.g. Bladeforged paladin 1 / sorcerer 14: max STR, then CHA and CON. [Final build sorc 18 / pal 2 to add Divine Might.] Tier-5 Eldritch Knight, BF Recon SLA, leftover APs into Earth Savant. I'd be curious as to how well it pans out for you.
EDIT: it's a question of priorities and expectations. If you think of this as a highly self-sufficient melee DPS build with WF/BF immunities and your no-save acid spells are just to proc EK bonuses, you'll probably have better results than if you expect this to be melee/caster DPS hybrid.