In many ways the sound you create through using effects can become somewhat of an extension of the instrument itself. Dirt is a great example, but delay and modulation effects can also have an influence on the performance itself. Distortion (or fuzz or overdrive if you like) is a bit more critical, imo. Your playing dynamics and how you allow the notes to sustain while you're playing will change depending on the gain structure of your sound. I think it's critical to have that sound in the right ballpark to be able to pull off a performance with the right feel. There's a feedback loop between your brain and the guitar, amp and any effects that you want in the chain that affects what your fingers do with the performance.
If you wanted to have something you could use to reamp, I guess you could use a DI box to split the signal so you have something dry with no effects on it to use. But then you'd probably still want to have a conventional amp set up to monitor through while you play, so you have a sound. It's rare for someone to just plug into a recording console and print that sound. It's not a conventional sound. I say rare because sometimes people do it, but I think it's usually to get that unconventional sound, not to turn it into something different after the fact. If there's no amp (or profiler, modeler, IRs, effects chain etc...) to monitor when you print the track, it's a very different feel.
Another consideration would be the order of your effects chain. Anything that you would typically use in an effects loop might be safe enough to apply after the fact. If the effect is wah or univibe, or something that typically wants to see your guitar's input signal before the rest of your pedal chain in front of the amp, it's not going to sound the same after the fact.