Yes, you are right. The Microchip part is just the USB PHY and with no logic related to PCIe. If you want a full stack USB host to plug into the PCIe slot of a Windows or Linux PC, the cheapest silicon I can find is the $4.38 ASM1042 on ebay. The TI parts are about twice that. For the USB device controller, the only PCIe to USB device controller I can find on the Market is the PLX (Broadcom) part (which is probably why they charge so much for it). FTDI has some USB device controllers for ~$4, but they are UART to USB or SPI to USB.
Also, Monkeh's comment about the final cost being more than the silicon cost is valid because of the requirement of the PCIe driver. ASMedia has a free driver, but it is most likely specific to their board and without sources. If it works you will not be able to modify it. If it doesn't work you will need to become a PCIe driver expert and write one.
If this is a project where you are trying to learn what it takes to build a PCIe device to USB host controller, I would suggest a Xilinx Kintex 7 board with PCIe connector. These are not cheap but you have free Xilinx PCIe and USB IP and drivers when you buy a Vivado license. Also you have all the sources and visibility so you can learn and modify it. As Monkeh mentions, the engineering here is not trivial and will cost you more than a few dollars. Over time this price will be minimized to it's lowest manufacturable cost given market demand for it.
My observation about standard product ASICs is that you must use them as the vendor intends. I am not aware of any ASIC vendor that gives away the sources to their PCIe drivers, though if you buy enough parts from them they might. The driver is part of the vendor's NRE just as the cost to fab the silicon is.