Based on the available evidence at this point, the recruitment panel headed by John Bercow (he wasn't making the decision alone) picked Carol Mills as the best candidate for the "Commons chief executive" part of the job of Clerk of the House of Commons, as a senior manager, in overall charge of the various permanent staff who keep the House of Commons running. With regard to this, Mrs Mills' experience as the secretary of the Australian department of parliamentary services was entirely relevant and appropriate.
Mr Bercow later went on record as being prepared to split the job in two to separate the constitutional role (where Mrs Mills CV did not show a track record in drafting legislation and understanding of UK constitutional law) and the managerial one. A House of Commons committee on the subject recommended "The Clerk of the House should remain Head of the House service, appointed by Letters Patent, but should not also be titled Chief Executive" instead "a new post of Director General of the House of Commons should be created, reporting to the Clerk but with clearly delineated autonomous responsibilities for the delivery of services." The eventual appointment of the previous deputy to the role ended the debate on the role of the Clerk. The new role was indeed created and is currently filled by Ian Ailles.