It depends on where you're recording. Every reputable studio where I've recorded has had several snares and cymbals on hand if I've wanted to incorporate a sound I haven't brought in with me. In terms of snares, however, I find it just as easy to adjust the tuning/muffling of a snare that's already mic'd up as it is to switch to different snares between tracks. Efficiency is key in the studio, as the clock is always running. Things move quickly. Laboring over gear selections can be costly.
I have never, under any circumstances, been asked by an engineer, a musician, or a singer to use gear other other than that I've shown up with. No one has ever said, "Oh, your wood snare doesn't fit for these tracks. You should have brought a brass shell." That's mostly forum talk. Real-life playing doesn't exhibit those expectations, at least not in my experience. Others may have different accounts.
As for toms and bass drums, I've never taken extras to a session. That would be a logistical nightmare for the type of recording I do. There's just no time to fool around with complicated gear substitutions once things get started. Clients want me in and out as quickly as possible. They pay by the hour and generally don't appreciate gingerly approaches. I set up, get mic'd and sound-checked (which happens very quickly), and start laying down tracks, which I've usually prepared for by practicing to rough tracks, otherwise known as scratch tracks. Often, I do two takes of each track, just for comparative purposes. Then I pack up and hit the road. It's a wham-bam process that isn't half as painstaking or romantic as some may imagine.
I should point out that I don't record with Taylor Swift or Justin Timberlake, stars who enjoy generous budgets and who might work on records for months at a time. I do mostly demo stuff for singers/songwriters and for local bands that are between drummers. Consistent, reliable, and efficient drumming is what matters in my case. No one cares how much gear I have, what that gear is composed of, or how that gear is constructed. I've never gotten a single session because of my badges or labels, and no one cares if my shells are metal, maple, or mahogany.