It is not lazy to try to control population growth, it is the only viable longterm solution because technology can not solve this problem. The goal must be to stop population growth or, for practical matters, increase the doubling time at least to the order of millennia. That is why I gave the example with colonizing the Milky Way, to make clear how bad the problem is and how infeasible technological solutions are.
less desirable options: uncontrolled global civilisation collapse with rapid decline in living standards and population, nuclear war, etc
more desirable options: intentionally adjusting lifestyle / affluence in western countries, non-coercive methods of population control (e.g. better healthcare, increased access to family planning, increased opportunities for women to participate in education / economic activity, etc), less focus on short-term appropriation of wealth
One could ponder whether it would make sense to use population control in addition to improved technology and development to make the ride across the world population peak smoother, to make the peak lower and happen earlier. There would almost certainly be at least some benefits to this, but given how long political debates about that would probably take, never mind implementing and enforcing the decisions, there might actually not be enough time to do this and get some noticeable impact out of it.
As long as it looks like the world population will peak around the end of the century, go for technology to make it as pleasant as possible, but if the outlook indicates that the world population does not peak or at least stabilize in the foreseeable future, then population control is really the only way or it will become pretty unpleasant as you indicated in your first scenario.
Is there a recognized exhaustable availability of resource X? Then we simply don't allow that resource to be used as a means of capitalist value extraction from a planet. The resources belong to the commons, and we must shift away from allowing profits to guide the use of raw materials.
I look at, say, the last century of the automobile, and cannot avoid wondering just how different our lives would be if we'd bypassed the idea of placing a couple tons of metals and plastics in everyone's driveways. What else might we have used those materials for, if we mandated that using them had to be done in a way that benefited a large number of people with the same material cost. This really goes for nearly every consumer product made available to the markets.
I don't think we'll be able to come up with viable long-term strategies for handling population growth, and providing a high quality of life to all members of that population, until we move beyond status-driven consumerism and the capitalist economies that create and depend on them.
There is a growing group of people who believe the world is over populated. I want to push back against that mentality a little bit, because we should not resign ourselves to hopeless thinking. There are plenty of things we can do to improve our situation before resorting to dramatic actions. I want to avoid "there is no other solution" like thinking. Once we stop looking for other solutions, it will become a self-fulfilling prophecy. That is what I mean by lazy and morally hazardous.