The fact is that even if Frida remained true to the Ford template her detractors would have even more ammunition, comparing and complaining.
No EXISTING HOUSE HIRES A DESIGNER TO REPLICATE WHAT PRECEDED THEM! They hire people with their own vision. The suits decide if that is best for them or not.
Does anyone think Lagerfeld' Chloe for example was the same as Phoebe's or Andersson's?
Of course not, but both Lagerfeld's and Philo's aesthetics, although different, worked well with the existing Chloe aesthetic and ethos.
As I've said before, I don't think it's, generally, the case that people expect, or even want, designers of a fashion house to replicate the outgoing designer's work; it's more a question of expecting them to be sympathetic to the enduring aesthetic, or 'undercurrent' (as Jungla_Juana aptly put it).
Both Lagerfeld and Philo were very sympathetic to that 'undercurrent' (Lagerfeld said that he showed his softer side at Chloe and Philo paid heed to Ralph's
advice that; 'people come to buy at Chloé for what it is, so it does not make sense to order a Chinese and get delivered an Indian!').
Incidentally, Philo produced quite a few designs that were obviously influenced by Lagerfeld's earlier designs, patterns, or motifs; just with her own spin put on them and as we know, she was very successful at Chloe and showed no sign of being up for the chop before she left of her own accord. So whilst it is true that; 'No EXISTING HOUSE HIRES A DESIGNER TO REPLICATE WHAT PRECEDED THEM!', it is also true that design houses
will often hire designers to rework successful older designs, patterns and/or motifs and it is then down to whether the designer's own aesthetic meshes well with these older designs/patterns/motifs, or not, as to whether that reworking is successful (in the the case of Philo), or not (in the case of Melim Andersson [IMO]).