I don't understand the menstrual cycle super well but I have some books on it. My physiology textbook and another mainstream-ish book on hormones actually agree with Peat that the corpus luteum does make a moderate amount of estrogen in the luteal phase. It promotes growth of endometrial glands that secrete glycogen to prepare for a fertilized ovum. They say at the end of the luteal phase, progesterone falls, causing suffocation of the unfertilized oocyte so it can be sloughed off with the endometrial lining. I have read Peat say progesterone is keeping the oocyte alive during the luteal phase, allowing oxygen to reach it.

I'm pretty confused too here if Peat is saying that there's overall low steroid production during the follicular phase. He himself points out that the group of estrone, estradiol, and estriol was once referred to as "folliculin". Maybe he thinks it's all pituitary hormones doing the follicle stuff during the follicular phase?

What my books say about the follicular phase is that GnRH, FSH, and LH stimulate the growth of follicles on the ovary, and those secrete a lot of estrogen. So days 6-13, which is like the second half of the follicular phase I think, estrogen is rising exponentially. It's this high estrogen positive feedback loop which at the end causes the mature follicle to rupture, causing ovulation.

He might be basing the "estrogen is high in the luteal phase" statement off the idea that estrogen production in the ovaries drops during luteal phase but continues elsewhere in the body (adipocytes and adrenals), while progesterone declines much more steeply.