Trying to get a 32.768 kHz crystal to oscillate

Rereading, I'm pretty sure I need 25pF caps instead as I misinterpreted how load capacitance worked.

You need 25pF caps less any stray board and chip capacitance. You'll probably end up at around 18-22pF.

However, I don't know what else is wrong.

Using a 74S part is probably killing you, for starters, if you're really using it. And if you are, then R2 needs to be much lower than the megaohm range.

Switch over to a 74C04 or a 74HCU04 (the "U" is important -- it stands for "unbuffered", and means that there are fewer stages of amplification internal to the part -- this makes its action less positive as a gate, but more linear -- and you're using it as an amplifier here).

If you don't switch over to a 74C or 74HCU part, then start by choosing a value of R2 that puts the output at around 1.5 to 2V when R1 isn't present (VDD/2 for CMOS -- 1.5-2V is specific to TTL). I'm not sure what that value will be, but I suspect it'll be in the 10-50 kiloohm range. Then see if things work. Find the largest value of R1 that'll work, then cut it by about a factor of 2.

I suspect, however, that to make a 74S part oscillate you'll need to overdrive the crystal. The thing is designed to work on a whisper of power, but the 74S is a brute compared to a typical CMOS gate. I'm not sure if you'll break the crystal outright, but you may have problems with temperature and aging.

Moreover, I want to simulate it in LTspice but I don't know how to get the motional parameters as I only know the motional resistance.

Motational capacitance = teeny, motational inductance = ginormous. It really only matters if you're trying to characterize the thing for start-up time (it's the motational resistance that matters for whether it'll oscillate at all, and whether you're overdriving it). I'd just guess at a Q of between 10,000 and 100,000 and solve for the motational capacitance and inductance from that and the published motational resistance (it's effectively a series circuit, so use \$X_{Cm} = X_{Lm} = R_m Q\$).

If you just have to know, and if you have a good enough signal generator you can measure the parameters. It would be too much of a digression to explain my (quirky) methods here; just Google on "measuring crystal parameters" and choose a method that matches the equipment you have available.

74S04 is completely wrong chip for this and would require very different biasing to get it work. Usually a 74HCU04 or 4069UB is used with that kind of circuit.