Hi there! first off I just wanted to say you did a good job explaining your ideas. The train of thought you followed is actually similar to a though Einstein had. The logic you used is completely acceptable if Newtonian Physics were all there was. The fact that you arrived at this conclusion on your own is quite impressive! Most people don't like to think this deeply about a subject. To explain why we can't hit the speed of light I'll need to explain/establish a few ideas. For context, I'm a graduate student studying physics, but since I am a student, there is a very slight possibility for a degree of error in what I say.

Newtonian physics is great. In fact, for almost everything we do, we use Newtonian physics. NASA uses Newtonian physics to calculate flight paths, even car manufacturers use it to calculate the acceleration, speed, and range of their vehicles.
Einstein realized that Newtonian physics failed to describe a few things though.

So most of the time if two objects are traveling towards each other at speed v in one dimension we just add their speeds together and there you go. However, this isn't exactly accurate. Even for two objects traveling at 30m/s they don't actually have a collision speed of 60m/s. It's actually 59.99981988 m/s. This is found by using Special Relativity's Velocity addition formula. In which the two speeds are added, and then divided by