Would it be problematic to have a female Doctor?
No. But neither do you need one to enjoy what the show has to offer or to prove the show isn't promoting sexism.
I guess I just disagree that the lead women in the earlier seasons were somehow "stronger."
I am surprised criticism is prevalent. Maybe it has to do with its status as a popular family show? Where having perfectly "properly portrayed" characters or PC theme is most important? And given what's on TV, Doctor Who is very benign (if there's a theme or morality pushed, it's usually some variation of "it's good to help people even at personal cost"); it's partly why I watch it.
Maybe my threshold for boycotting certain media is just looser than yours, which is fine, too.
I haven't mentioned boycotting. Just concern.
Criticism seems not to be that the women have to be perfect or "strong". Far from it. It just seems that the vast, vast majority of Moffat's female characters always seem to end up being defined by either husbands or children, as if all women are simply destined for marrying and/or breeding. Real people are more complicated than that.
But it is a complicated issue and I am not familiar with the show, so it is difficult to discuss.
You really don't have a leg to stand on here. You're judging the guy as a sexist for things other people have said about him. Go look at the stuff yourself, then come back with an opinion.
I mean, seriously, let's go through his female Dr. Who characters:
The Empty Child/The Doctor Dances: Nancy - takes care of children in the London Blitz. Eh, it's somehow traditional, of course, but she does it in a strong way, and I have a problem with having a problem with women doing traditional things. Being forced to do them, sure. But doing them at all, no.
The Girl in the Fireplace: Madame de Pompadour is based on a historical figure. There are pros and cons, yes, but at least she takes a keen interest. I grant you that she is largely the one getting rescued, but again, historical figure.
Blink: Hard to find a better strong female character than Sally Sparrow, no? She is trespassing, breaking into places, adventuring, detectiving, working things out, saving the day. By far the protagonist of the episode, much more than even the Doctor.
Silence in the Library/Forest of the Dead: There are a few here. Ms. Evangelista is a bit of a ditz, but at least it's pointed out that she's not worthless, though other characters think that. Anita, she's like the pilot or something? She doesn't do that much, just a minor character, like either of the Daves. CAL is a child, and actually she's saved everyone. And ok, Professor River Song. She basically takes charge, well, along with the Doctor, and overrules him to make the self-sacrifice at the end. Later on, she is basically *always* taking charge of the situation.
Amy Pond: Also takes charge, to the point that the other character's are joked as being subservient to her, and her husband is Mr. Pond at the wedding, "Wait, that's not how it works... yeah, yeah it is."
I mean, it's not perfect, I grant, but it's not near the top half of problems I see towards women on TV.