My understanding is that they define gender with respect to historical oppressions which are imposed on people according to their sex.

Yes, and no. They are doing this AND assuming that transgender women do not suffer misogyny. They do, oh yes, they do. Ok, I could kind of get it if they were like "5 years of suffering from misogyny before you can be considered a woman". But no, they just deny any trans woman at all from the category of female.

The goal is to remove these oppressions and stereotypes entirely. This doesn't make sense from the trans definition of gender which requires an innate masculinity and femininity and an individualistic version of gender.

Actually, these two ideas are not mutually exclusive. You can believe in the idea of innate gender (and not just along the binary) & innate masculinity/femininity AND also believe that oppression & stereotyping is bad. Yes, you are right, the transgender view of gender is problematic for those who want to argue that gender identity is entirely a social construct (generally for the purposes of removing the idea of a physical inferiority for women). But that's a niche case. In general, there is no conflict between believing in an innate (or at least brain-structure determined) gender & feminist opposition to oppression.

I'd note in particular that I used to think transgender experience was an argument against Judith Butler's groundbreaking work. But Judith Butler herself in her statements on the legitimacy of transgender experience convinced me otherwise.

but society decides how it treats you based on its declaration of your race and gender, not your individual identification

A big part of the reason behind the concept of "passing" within transgender circles. (Passing = when people you meet assume you are the gender you identify as without noticing you're transgender.) That's also frequently the point where transgender women note a sudden increase in misogyny. Listening to transgender people (men and women) explain just how extreme ordinary, every day sex discrimination manifests from both perspectives is pretty interesting, actually.

Therefore, as it concerns oppression and alleviating oppression, what matter is how society defines you not how you define yourself.

In essence, the kind of feminist you are talking about gives the majority the right to determine who is a woman and who is not - and this same majority think some trans women are women and some trans women are not women. It's a pretty silly system to adopt for personal morality & action, in my opinion. Why go with the majority opinion? Shouldn't feminists hold themselves to a higher standard? That is, shouldn't they treat others based on how they themselves, as feminists, see others, not how society at large sees them?

Also, I would add that point of view tends to minimize or ignore the effects of intersectionality. If a certain social experience is needed to be a woman, how do you deal with radically different social experiences? Are one of 1) Mozambican women and 2) Canadian women more womanly than the other, because their social experiences are so different? And if Mozambican women and Canadian women can both be just as much women, or just different types of women, despite their different social experiences, why can't trans women just be a different type of women, despite their different social experiences?

In general radical feminists don't dislike transpeople

That is my experience as well.

there isn't any "transphobia"

That is not my experience, unfortunately. While feminists do better than the general population, in my experience there are still some feminists who find transgender people "icky". As an example of this, search "transgender" at /r/gendercynical . As you note, however, bad apples, not a bad batch.

In a world without gender as radical feminists define it, people would be free to express their personalities without the harmful backdrop of stereotypes and labels.

I agree, and as a transgender woman I can tell you that I personally cringe when some other transgender women use an overwhelmingly "pink" (stereotypical) presentation to insist on their gender identity.

I disagree, however, that gender identity (as opposed to socially defined roles) can be entirely divorced from biological sex (specifically, brain structure). If nothing else, I've noticed that some gender dypshoria (the discomfort in the mismatch between body & identity in transgender people) results from a sense of not seeing one's self as healthy. The same way someone sick or starving looks 'wrong', a transgender person will see incorrect physical reality (beard, lack of breasts, lack of testosterone-fueled musculature for trans men, etc.) as 'wrong'. While I generally agree with Judith Butler's theories (though not necessarily with the theorists that build on her work), I maintain that gender identity is biologically driven at least to that extent.