Oh man...here I go.
For interstellar travel to matter several criteria need to be met.
1. Some form of stasis must exist to allow the same people to start and end the trip in relatively the same physical condition.
Relativistic effects make this a given, if the craft can reach sufficient, less than light-speed, velocity.
2. The traveler must have a destination worth traveling to.
I think here you forget that it is the nature of some people to explore. We've been doing it for generations.
3. Somebody need to be willing to BUILD these ships.
The cost aspect is a major point. However, it's only a show stopper if you think of your first destination as being the stars. Consider instead a progression, with increasingly high-performing ships being used to travel out own solar system, in time becoming as mundane as aircraft are today. Then you might start to explore (and exploit) the outer reaches of the solar system. With that kind of progress from today, humanity is that much closer to interstellar voyages. And indeed, though I envisage us travelling the galaxy (if we don't destroy our civilization and go extinct), a journey of a million light-years must begin with a single step.
If somebody actually wants to reach their destination in a reasonable amount of time, they would more than likely choose to sustain several g's of thrust throughout the entire voyage, or at least until they reached 80% or so of light speed (which would take several decades of sustained thrust).
One g is probably a good thrust to use. That gives you an artificial gravity equal to that of Earth, which is nice to have. You could run less if it makes things cheaper or easier, but you wouldn't want to run much more or daily activity on board the ship would get difficult.
We may want, or even need, to find a new inhabitable planet, but have you considered that even in our most dire need, human ingenuity shall eventually fail us, causing massive death or extinction as a species? We are but one incredibly small speck of dust on the timeline of the Earth, let alone the universe. I believe that our time will come to end, and that will simply be it.
Oh I agree, this is the more probable outcome. But that doesn't mean we should not discuss space travel; quite the opposite.
If you're actually racing, you have a bit more of a problem since you will only be able to accelerate up for the first half of your journey (You'd need to accelerate down again in order to stop when you reach your destination).
This applies to an extent no matter what. Even if you coast for some of the journey, you need as much deceleration at the destination as you need acceleration at the start, unless you do gain a significant amount from ISM drag.
Since carrying the extra mass of fuel out of Earth's gravity well actually prevents you from being light enough to escape without dumping the excess containers, the race methodology will probably always be unfeasible short of some superefficient engine that yet does not exist.
You don't build your interstellar spaceship in low earth orbit! You'd probably want to build the spacecraft far from the sun, and not orbiting any planets or satellites, unless you do plan on using boosters. I guess you have a trade-off - the closer you build to your raw materials, the less fuel you require shipping them from source to spaceshipyard, but then the more work the ship itself needs to do. You'd want to get your materials from somewhere without a deep gravity well. Outer moons of Jupiter or Saturn, perhaps, though they are rather small. Or the asteroid belt. Or perhaps even the Kuiper belt.
Or the likely event that Earth's Moon will be the most developed and industrialised solar system body after Earth itself may make that the only place where sufficient materials, technology, and expertise can be had to build the interstellar spacecraft, making the construction location lunar orbit or a Lagrange point.