The Falcon has a very interesting connector: the DSP connector - it has SPI-like communication with DMA to use quite a high throughput and a programmable clock divider with support for external clock if the internal is not suitable.
So naturally gives the idea Why not connect an SD Card in SPI mode to it?
Electrical interface:
The SD card runs on 3.3V but the DSP port only have a +12V Vcc, so needs some better 3.3V regulator that can handle input up to 15-20V (like AMS1117 3.3 or LM1117 3.3 ). Next is to connect the signals a regular 74HC244 can be used (powered from the 3.3V).
There is a bit other difference: the SD card uses the rising edge of the clock to latch but the DSP port (playclock/recclock) uses the falling edge, so it needs a NOT gate.
Signaling:
The play-/recdata can be the MOSI / MISO. Both SPI and this serial port uses MSB first, so no need to change order.
The playclk can be inverted (rising edge instead of falling one) and used as SCLK (do not connect with the recclk as both are output!)
And finally some special modes. It looks the best way to use the serial in/out is in handshake mode to be synchronized with each other (both starts the same time), but that needs an additional signal. There are 3 GPIO, ex. GP0 can be connected to both SYNC and used to start the transfer.
The SD card still need another signal, the SS (slave select), that is sadly active low, so maybe GP1 could be used for this purpose (or maybe just negate the SYNC and use just that)
Then there are a few problems/uncertainties that needs to be addressed before:
The transfer is based on 2byte words while the SPI is byte based, but doesn't looks that important as the commands are 6bytes long and then read/write is usually 512bytes. Just in some cases when waiting for R1 response does it matter, but that can be done by checking both halves of a word.
And this is the current end of my knowledge.
There are the following questions I need to be answered:
How to check if a serial data has been transferred?
In the Developer manual there is the line: "A minimum of two clock periods will always be inserted between data words."
What does it mean?? Can't produce gapless datatastream? Then how does the diag cart can check the serial in-out with just wire-loopback? Or is it just mean that it needs time to set up again but not the actual signals produced on the output?

It may be easy to test, but i don't have the time and knowledge to do by myself...
Then it just needs a driver/program that can init an SD card (preferably with support to SDHC and SDXC) -- don't forget to do it with a clock slower than 400kHz, then it can be ramped up to 32MHz/4 (that's the smallest clk divider), and just need a FAT interface to be included.
For skeleton I think the hxc_sdcp source could be used, it already have some browser-like feature, and a FAT32 library, and available here: http://onorisoft.free.fr/retro.htm?atari/falcon.htm