It has organization and technology. They wouldn't need to kill off eliminster just to shepherd the primitives into the modern age, establish proper trade communication networks and so on. Even something as basic as mining. Forestry/logging, mining, etc would be upended to the point of the old ways collapsing and becoming unsustainable as modern methods are introduced to those willing to sign on the gas line of the contract with the new colonial power.
As to those high level wizards, the dragon mark houses have plenty of resources to hire them and ways to offer them a better life.
Apologies, but you're mistaking the power of high level wizards.
Eberron has NOTHING that the wizards of Oerth and especially FR can't duplicate, since the number of high level Mages vastly, vastly, VASTLY outnumbers that of Eberron. Every major city has a mage of 15+ toodling around, all the larger ones have an archcaster of SOME kind, and the BIG cities have multiple archmages.
Eberron has exactly 1 Druid/20, the highest caster in the place, and he's a pine tree.
If Eberron goes to Faerun, some high level character are going to come through, loot the place of its knowledge, eberron will find it almost impossible to stop them, and then they'll all go home laughing.
What the Red Wizards of Thay, who average level 11, could do to Eberron does not need to be described. The average Red Wizard is equal to a great eberonian hero, and the Zulkirs are post-20's with NO EBERRONIAN RIVALS.
Hells, FR has a mercenary company whose core power are f/6's!
Faerunians are used to dealing with dragonflights and renegade dragons. Eberron runs screaming from dragons. Deities and demigods seem to be running around all the time, what's this Lords of Dust and mutant guys underground? that's like, great! Great xp, man!
Seriously, the smart eberronians would very quietly show up, realize there's the equivalent of an guildmaster wizard on every other street corner, with itchy fireball fingers, and go home before some decide to emigrate and take them over.
Elminster? Yeah, if he was serious, no Eberonian team would ever get a toehold on Faerun, simply because he could unequivocally shut down permanently the transfer method, AND remove permanently anyone who could replace or fix it. You little artificer/12's should not mess with Wizard/29's, you know?