Unfortunately, there seems to be a significant amount of missing information in the OPs thought process about using carbide tools to cut through armor. A carbide tipped/toothed sword might look cool, but I seriously doubt it would work very well.
Disclaimer: I'm not a professional machinist, I only pretend to be one (sometimes) at night and on weekends. I don't have all the answers, but I know of plenty of problems.
Feeds and Speeds
When doing milling, cutting threads, or anything to do with machining metals, feeds and speeds are absolutely critical to the process. These calculations take into account the type of material you are cutting (by actually knowing what it is), the cutting speed (as accurately as possible), the size of the cutter, and even the angle of the cutting blade to the surface (baked into the type of machining as well as the tool used).
If you get it wrong, you risk damaging the tool to the point where it dulls or even breaks. With many machining tools, it won't even cut below a certain speed, it just wears the tool out prematurely by grinding the material away instead of cutting.
This is a science and machinists know it can go very wrong very quickly. "Wrong" being a damaged or completely broken tool. There is definitely a fair amount of "wiggle room" here, depending on your machine and tools, but you won't know how much you can "wiggle" until you are familiar with the tool, machine, and the material. With the material changing with every single opponent's armor, you're never going to be able to know what's safe until it's likely too late.
Without knowing the exact material and hardness of the material you are up against, you immediately risk damaging your sword. Since no human can gauge their swing against a mobile attacker to inches per second or feet per second until it's already in motion and not with the accuracy needed, there's no way to know if it's appropriate to actually cutting the armor vs blunt force. There's also no way to get the correct cutting angle between the sword and the armor, since both are in motion and a change in surface curvature in the armor might cause the carbide to skip off, rather than cut. Since carbide is so brittle, skipping off amounts to a hard shock which can break it. Even sword made of steel can't be brittle, since they need to flex with every blow the take or give.
Even if you had a motorized chainsaw sword, you still aren't getting the feeds & speeds right, so instead of breaking off 1-2 teeth, you're breaking off 10-20, if not all of them.
Milling processes need to be done with a specific cutting depth and chip removal. If you remove too much material at once, you also risk damaging the tool. If you don't remove the material fast enough, you risk cutting it again, preventing you from making a deeper cut in the armor.
This is managed partly by the tool as well as by the feeds & speeds mentioned above.
If you don't have the right pressure between the tool and the work piece, you risk breaking the tool. This is why machining machines usually have really tight tolerances and machinists talk about backlash (recoil arising between parts of a mechanism) and chatter (to vibrate rapidly in cutting) quite a bit, since they can negatively affect not only a cut but also damage tools. Both can cause excessive tool wear as well as not correctly cutting the material.
When using a tool freehand (sword) with a material that isn't in a vise or otherwise clamped down (opponent), you aren't going to get the correct tool pressure to get a good cut. Depending on circumstances, you may have too little pressure or too much. Too little and you don't get a cut, and too much damages the tool.
You also have to worry about how far apart and the size of your teeth. Uncommon knowledge says that you should have at least 3 teeth engaged with material, or you risk shearing them off. This goes for band saw blades made of spring steel, but I can imagine it works for just about anything.
Also, if your teeth are too close, you'll need more pressure to do more than scrape across the surface. Too far apart and you have too much pressure. If the spaces between the teeth are too shallow, they clog and prevent you from removing more material. However if they are too deep, you risk compromising the structural integrity of the teeth.
Again, this all depends on the material you are cutting as well as the speed and feeds you are trying to cut at.
Carbide also comes in inserts for tools, so when they get dull, you can swap them out easily and either sharpen or toss them. This is for mid to high grade milling tools, and they can cost quite a bit. This allows for better tool maintenance and longevity, since you can replace a single tooth, instead of scrapping the whole thing, or having to to a major repair in the field.
Replacing a broken tooth still isn't something you're likely to do in the middle of a battle, but rather something you'd do before/after a battle, since it still takes tools and time. You also don't want to be carrying around a 20 lbs. bag of replacements that can get in the way, wear you down, get sliced open, get cut off, or otherwise inhibit the battle.
With all the things mentioned on this thread about carbide tools, it just doesn't seem to be a good idea for use with a sword. There's too many things that can go wrong and other materials that are better suited, as mentioned in other Answers.
Some of it can be mitigated with the carbide inserts, but not enough to make it more effective than the current steel used in swords. Adding the complexity of all the different considerations with speeds, feeds, tooth distance, tooth size, and more, it make this nearly an impossibly unsolvable math problem to avoid completely damaging your sword beyond use in a single encounter, let alone multiple opponents on a battlefield. This might work, marginally, in a duel, but not when potentially engaging with multiple enemy on a battlefield. If your sword isn't reliable enough, then it's basically useless for more than blunt trauma, which makes a bad sword.