The flaw is that you don't gain mass as you go faster. You gain momentum. That momentum energy even causes gravity in a way[1].

The reason it seems to take infinite energy to go faster than light is that the rate of time for the traveler changes as they go faster. From your observer's perspective, whenever they accelerate the speed increase is exactly cancelled out by the change in relative time rates. The more energy they spend the faster they go, but never quite fast enough to go at C.

From the traveler's perspective it is a totally different adventure. When they add energy they accelerate. Period. They don't notice the time dilation outside of observations of their origin and destination. If they have enough fuel, they can accelerate smoothly to 2 times the speed of light and travel a light year in 6 months. The problem is that more than a year will have gone by in the origin and destination on the way and they were never observed going faster than C.

If we had a way around the rocket equation we could easily cross the Galaxy within a human lifetime. It would just be Earth would be gone by the time we got there.

[1]https://physics.stackexchange.com/questions/63961/does-relat...