The Raspberry Pi Pico is good for a lot of things; this is not one of them (due to the lack of at least USB 2.0 High Speed). An LPC18xx, LPC43xx or even IMXRT106x would've been a much better choice.
You might only be able to get 8-16MB/s max out of some of those old drives (especially pre-1994), so you probably aren't missing much.
edit: whoops, that's bytes not bits per second
or good oldshool USB 2.0 Cypress FX2LP CY7C68013A or its Chinese clone Corebai CBM9002A
cy4611b___usb_2_0_usb_to_ata_reference_design https://www.farnell.com/datasheets/62992.pdf Back around 2005 a lot of USB HDD enclosures used this exact chip.
All the above needs is LBA to CHS translation.
It's a question of what tradeoffs you're willing to make. If you're making a professional product then sure, but I've checked the chips you suggested and the cheapest one available on JLCPCB seems to be the LPC1820FB at $6.5. If you fuck up a revision or two at 5 pieces each that gets expensive rather quickly for a hobby project.
It's a shame really that ULPI is such a complicated interface, at least compared to RMII or SDIO, otherwise you could just buy a high speed USB ULPI phy and use it with the RP2350 via PIO. Eben even mentioned at some point someone working on a PIO ULPI implementation, but I'm assuming that went nowhere because they couldn't make it work reliably.
An alternative approach I've been considering is using a ch32v305 as a "smart" USB-HS bridge, and connecting the RP2350 via SDIO to it. The problem with that of course is you have to implement most of the USB stack on the ch32v305, and the documentation on those chips isn't great to put it mildly.