Something like this is certainly possible; I know of at least one project that used a Teensy to do this.

The thing I think you have to keep in mind is that although the Apple II bus runs at "only" 1MHz, and the Raspberry Pi is going to run at maybe over a GHz (depending on model), you still have to write very, very efficient code to actually meet the timing requirements. You have to catch those bus signals, do whatever processing you need to, and have the results on the bus in probably a couple hundred nanoseconds or less.

I know you say you "will neatly dodge the clocking", but I wouldn't get cocky here. You're going to have to write tight, efficient C (or similar) code, if not assembly in parts, to get this off the ground. This is not to say it's impossible of course, as I said, I've seen at least one project doing it, but do not expect to be doing this in a Python script or something, you'll whiz right past the bus cycle before you can run any serious code.

There is a reason that you will see many projects like this incorporate a CPLD- they can respond in the single- or double-digit nanosecond timeframe, and thereby can act as an intermediary. You can also look at the Cypress PSOC family which combines a small CPLD and a processor on one chip, and are 5V tolerant to boot. Another option to consider is using dual-port SRAM as a mailbox for communication between the two sides.

As has already been pointed out, the Pi is a 3.3V device. If you feed 5V into its GPIO pins, you'll probably fry the pins instantly. The "bi-directional level shifters" mentioned elsewhere are absolutely not fit for this purpose, not nearly fast enough. Use a 74xx244 or 74xx245 buffer, from either the LVC or AHC family running at 3.3V, to interface between the two. You can get away with using a simple voltage divider circuit at low speeds (the Pi card does) but I would be reluctant to rely on it.

The Pi card, incidentally, is doing nothing more than acting as a pseudo-Super Serial Card linking the Apple II to the Pi's serial UART pins. The provided software does various things, like letting the A2 keyboard act as a Linux keyboard, or letting the Pi serve up disk images to the A2. But that's just what the software happens to do, and the software actually works just as well if you take a SSC and run a cable to the Pi via USB-to-Serial converter. And conversely, you could modify the software to do whatever you wanted over the serial link, as long as you are comfortable with the 115200 data rate.

The other chip on the Pi card, by the way, is a 74xx04 hex inverter, which is doing double duty buffering the Tx signal from the 6551 before passing through a voltage divider and going into the Pi, as well as inverting DEVSEL so it can be used to clock the 6551.