Today the power that people like Anna or Carine had is still there but on the lower scale. Delphine Arnault has proven that she had an eye for talents so she is the one making decisions. At KERING they still might go for head hunters tho.
I think it depends on a lot of factors. It depends on the CEO,
the board, the chairman of the board, the parent company managing executives, who the owners are, who the owners hired as a consultant to tell them who to hire, whether or not they have someone responsible for talent acquisition in-house (many do), or if they have that service outsourced on retainer (many do, too).
If you're a board, private investment firm, owner, or CEO and you have to hire a new creative or artistic director, you're going to use every resource available to you to make the best decision and leverage the best talent you can get. One that fits your strategy and your business challenges.
Stylists and fashion editors in general have a long history of playing matchmaker. Not just because they select the clothes that get shot, but because they've worked with a bunch of houses and designers and
they know who's working where. Oh, Schiaparelli is looking for a knitwear designer? Turns out Marie Chaix knows a guy who's looking to leave Carolina Herrera. (I made that up to illustrate the idea, that is not real, don't worry Wes).
Anna Wintour is the ultimate matchmaker, not just because if she throws her support behind a designer there's a good likely hood that they will succeed (but not guaranteed), but because
she knows everyone. And she knows her reader, who is also probably the customer they're trying to reach. And if she doesn't know, one of her editors will.
If you're hiring, you're going to want to let Anna know if she doesn't already let you know. It doesn't mean that everyone is going to listen to her. Many don't, but if Anna Wintour or Carine make a recommendation, you're going to check it out. Her power in that regard is definitely waning.
Most likely everyone uses ALL of the above means to create a shortlist of potential candidates.
Most brands will have the designers do a test project or a pitch to get the job. This is standard practice, even when you get to the big leagues. For every hire, there are probably 2-5 other designers who also submitted a project or pitch but were rejected. Maybe someone Anna recommended was good but their vision didn't work with the brand's product strategy. Maybe someone Panos Yipanis suggested was great but didn't have enough managerial experience, or didn't have any marketing savvy. Who knows. It's all case by case with a ton of different variables that we are blissfully ignorant of.
Also, many times designers are approached for a job but turn them down. I can tell you that Christophe Lemaire was asked to do Celine after Philo left and he immediately refused.
I can also tell you that Virginie Viard was not Chanel's first pick to replace Karl and that they had at least one designer waiting in the wings ready to go.
And I can also tell you that Paul Andrew, who left Ferragamo, was an Anna Wintour mistake---one that I would not have made.