Yeah for these kind of drums, and I assume you mean the bassline as well, distortion is your friend, it really doesn't sound like anything fancy.
You could get some mileage out of using simple clippers, like the fruity soft clipper.. seriously, it's pretty good.
There's many more ways to get it.. and better sounding. Tbh I know it's the style or whatever, but the distortion sound kinda cheap to me
The whole idea with those VST's that model old analog EQ's and comps is that they add that distortion and harmonics.
Guitar effects and amps are an old go-to, and again a whole range of VST's that simulate it. Waves and NI have a bunch, pricey though.
Lots of free ones too... just google 'free saturation vst' or 'free tape vst' and just check out a bunch of them to hear what they do.
It's pretty impossible to describe what tape saturation does versus tube saturation, or clipping.
Usually you'll want to add distortion fairly early in the chain because it'll almost always introduce artifacts or frequencies you don't want.
Definitely check if you're not adding a lot of subharmonic mud and remove it with a high pass filter. Filtering works better than EQ'ing here.
If you want to have the grit of the distortion without losing the weight and dynamics of the original sound, use the distortion in parallel. I would
probably split the sound across two mixer tracks: one clean and one distorted. That way you control the precise balance.
Another thing you might want to look into is using gates to make sure every bass sound and kick is completely killed before the next one comes in.
It's a bit tricky to get a hang of balancing it, but it'll give you a lot more control than sidechaining compressors.
Secret 'shit I only have a sampler' trick from the 90's: render your drums and downsample them in a simple audio editor. All the old sampler machines
people used back then (and people now still try to sound like) had pretty low quality sound. It would be like 33.1khz/12 bit or worse... it would have a good filter
and some effects to clean up the sound. Everytime you resampled something, you'd lose a bit of definition and add some kind of distortion..