Let me paraphrase the question to make it clear what I'm answering. The equivalence principle tells us that acceleration and gravity are (locally) equivalent. But everyday experience tells us that an accelerating object speeds off into the distance at an ever increasing velocity, while the $1g$ gravitational acceleration I'm feeling right now hasn't increased my velocity at all in the 54 years I've been experiencing it. So how can acceleration and gravity be equivalent?
The answer is that acceleration is measured relative to a freely falling observer i.e. an observer who isn't accelerating. If you are aboard the rocket accelerating at $1g$ then in your own coordinate frame you are at rest and your own velocity isn't changing. However if you throw one of your crew mates out of the air lock you'll see them accelerate away from you at $1g$. Your velocity is changing relative to your freely falling crew mate.
Likewise here I am on the surface of Earth experiencing an acceleration of $1g$, and in my own coordinate frame I am at rest and my velocity isn't changing. However if I throw you down a mine shaft I will see you accelerate away at $1g$. The gravitational acceleration is changing my velocity relative to you.
So the two scenarios are indeed equivalent. Sitting here at my laptop typing this I really am accelerating at $1g$, and as a result of this acceleration my velocity really is increasing relative to a freely falling observer.
But there remains a big difference. If my rocket has enough fuel I can carry on accelerating indefinitely and the unfortunate soul I threw out of the air lock will keep accelerating (relative to me) indefinitely. Hence as the question says:
If I was in a spaceship continually accelerating at $9.81m/s^2 =1g$ in a straight line, I would reach near light speed within a year.
By contrast if I throw you down a mineshaft you'll accelerate away at first, but only for a limited time. As you reach the centre of the Earth (it's a deep mineshaft) your acceleration relative to me will decrease to zero then change direction. If I wait long enough you'll start moving back towards me again. This is very different to the behaviour we see with an accelerating rocket.
The reason for the difference is that the equivalence principle tells us that acceleration and gravity are only equivalent locally. There are no completely uniform gravitational fields in nature - all gravitational fields change with distance. The equivalence principle applies only over a distance small enough that the gravitational field is approximately uniform. A fair comparison with the rocket would require the acceleration of the rocket to change in the same way the gravitational acceleration changed.