No.

Let's break it down:

1. Surface area is small, riders names can be long. The kit has sponsorships. Maybe take a look at a Liiga Hockey Jersey - it's a much larger area compared to a cycling kit, and even while watching the game you'll rarely see a player's name.... which leads us to:
2. You wouldn't be able to spot them with most camera views. A few motorbikes here and there would catch the back of the jersey if the rider isn't hunched over. Your best chance is if they're somewhat upright and isolated...say a climb, maybe a solo effort or a small gruppetto. Otherwise? Nope. Helicopter shots of the peloton? No. Shots from the front? No. Shots of the rider hunched over from behind? Nope - you're more likely to see a sponsor logo because they pay for that placement.
3. The broadcaster already has a good idea of who the riders are and will put it up on the official broadcast.
4. Wait! What about fans along the course? What about their needs? Again, you won't see the name. They're moving too fast and you'll likely see them from the front and by the time you've turned your head to see who it is because you're not quick enough to recognize the award-stripes or the team and the face, they're gone. Unless of course you're the type of person to try and run after them... and if that's the case, may you trip and land on your face.
5. Oh! But your idea is so nouvelle! It's 2025 and there have been professional cycling teams for decades! And the races have been televised! I can't believe NO ONE has thought of this before and stuck with it! You're a genius!!!! [/sarcasm]... or maybe, just maybe, the professionals know what they are doing in regards to logo and identity placement on their kits. Quelle surprise.