Unless the policy changed quite recently, I have never known this to be true. White bags do require a unique re-colouring process, but it’s not exclusive to one Spa visit— at least, that has been the case in the past. Additionally, I believe this pertains specifically to White bags* rather than all shades in the white colour family. While there may be an objective limit (in a pedantic sense), I’ve never heard of any limits set in place by Spa.
Regarding the example of the Craie Birkin, Hermès would be able to replace the handle if it were discoloured beyond simple cleaning or painting. You didn’t specify a leather type, but since the hypothetical bag is Craie, there should be no issues with leather availability, as all leather types used for Craie Birkins are still in production.
*—presumably including Blanc Cassé and New White but excluding off-whites like Craie, Nata, Poudre, Vanille, Ivoire, and Mushroom. I was told this is because the paint used for White bags contains a special compound in it to prevent yellowing. Not to mention, the white dye/paint Hermès uses has a nominal bit of blue tint to make it appear whiter/brighter, whereas off-whites are intentionally made with brown/yellow/pink undertones to appear less