I don't think so, personally I have found more emeralds while mining than ancient debris. However, I have not looked into the actual numbers, I may run some tests using WorldEdit to see what the rough rates are.
I should also mention that while I am sure you are referring to actual generation in the ground, emeralds are still fairly easy to obtain when you factor in trading and looting chests. I have only seen netherite in a chest once.
Emeralds are definitely rarer by a large margin since they can only be found in a single biome, which while fairly common is only a small fraction of the world, about 8% according to this thread; the game makes 3-8 attempts, averaging 5.5, to place a single block of emerald ore per chunk and if around 90% successfully generate (an estimate based on the percentage of blocks that are a stone variant) then there are about 5 emerald ore per chunk within Extreme Hills and 0.4 ore per chunk overall, which makes it about 4 times rarer than ancient debris, which averages about 1.65 ore per chunk across the entire Nether.
Notably, despite being less than half as common overall ancient debris is more common than diamond at its peak layers; diamond peaks at around 0.12% of blocks between layers 5-12 while ancient debris peaks at around 0.15% of blocks on layers 13-15, even higher on layer 14 due to its distribution (though in practice you'll expose at least 3 layers with the minimal 1x1 tunnel):
(for some reason I can't post the images directly)
For comparison, emerald ore (not shown above) has a concentration of around 0.07% (5 / (28 * 256), where 28 is the number of layers it generates over (4-31) and 256 is the number of blocks per layer), and since it generates as single blocks this is also the average amount found for a given exposed area (ores that come in veins are more likely to intersect a tunnel and will indirectly reveal more blocks after you mine the first one, increasing the yield per area exposed; this page claims that up to 1.7% of exposed blocks can be diamond ore, or about 14 times its per-block abundance, with ancient debris being maybe half as high due to its smaller vein size, which will effectively make emerald rarer even if it had the same per-block abundance as it can only benefit from the blocks exposed per block removed ratio).
This may change in 1.18 though; the Wiki says that there will be 50 attempts per chunk with a vein size of up to 10 (not sure exactly since they say 1 for Java and 0-10 for Bedrock; the latter suggests an average of 250 ore per chunk in mountains that are high enough, but much less in near sea level terrain, and also suggests that the new mountain biomes, with 20 attempts per chunk, are separate from Extreme Hills, so a larger percentage of the world will have emerald if the new mountains do not replace part of the original biomes, as e.g. Bamboo Forest did).