The inaccuracy is that in reality, the box of Legos is huge and nowhere near running out of bricks. If suddenly all humans on Earth stopped dying, the carbon cycle wouldn't even notice it for centuries or more (the things we'd be doing to control overpopulation are another topic though).
Why are you assuming the analogy is to physical resources instead of "population slots," for lack of a better term? The old have to die for the young to move forward, which is one thing that gives me pause about immortality research.
Because the analogy is about Lego box and making new houses, as opposed to e.g. fitting new houses on a finite Lego base plate[0].
I don't buy the "population slots". 100 years ago, the "population slots" were limited to at most 2 billion, back before Harber-Bosch[1]. There's a lot of space to use - we can live under the sea, we'll finally have an economic reason for that Mars colony. And we can stop making children so much too. Those things are actually easier than beating death, so - with added incentive to actually making them happen - you can expect them coming before people actually get immortal.
"The old have to die for the young to move forward" - This seems like a baseless assumption. If immortality research succeeds, surely we can also fix the problem of older people being stubborn.
A lot of the stubborness comes from facing death. When you think in terms of forever, it helps you to think a lot bigger than petty personal politics.
