Could it be that the "reputable aviation recruitment company" does not have a recruitment contract with EAAC??? These companies only make money from airlines & applicants willing to pay them for their services.
With a bit of groundwork it not difficult finding an airline job on your own providing that you fit the criteria. The worst of these agencies are the ones that charge silly money for so called cabin crew preparation courses.....I work with a stewardess who is a trainer on one of these courses & she is one of the least qualified to do so, knowing little about teamwork & has been knocked back for promotion on several occasions.
Confidence, hard work & the ability to be yourself are all thats needed to get a flying job.....avoid the sharks & good luck!