I've spent a good deal of time thinking about this and related problems. The puzzle caused by the multidimensional linkages between growth and technological progress has been a difficult one to crack. You can look up the giants in the field of growth, such as Solow, Ramsey, Simon, Harrod, as well as the New Growth Theory.
Production Possibilities Frontier
I have come to think that the best framework to start thinking about this issue is through the use of an expanding production possibilities frontier (also PPC).
I chose this because it illustrates beautifully the difference between efficiency improvements of existing technology (which, at best, can get you to the PPC) and the shift to new PPFs through the discovery and implementation of new technology.
Here's a practical example (notice the inverted Y cost scale):
General Purpose Technologies as Gamechangers
This should go without saying, but in reality, this frontier is a hypersphere. The Hypersphere of Human Possibility, if you want to sound a bit grandiose about it. Imagine new technologies as little pins that push outward at the frontier. Now most technologies (say the little cardboard coffee-holders at starbucks) are not that impactful, and only push the frontier outward a little locally. Other technologies such as the combustion engine, electricity or computers are different -- these technologies have a virtually limitless range of applicability - they can push the frontier outward in many directions at once, and by a lot.
Even more interestingly, one such gamechanger can open the path towards another. Combustion engines were used to generate cheap electricity, and eventually cheap electricity was used to construct delicate electrical equipment, such as computers. Computers then were used to design better combustion engines, power distribution grids, better computers etc.
Visibility of New Technological Possibilities
The fundamental question is whether there will ever be diminishing marginal returns to new technological research.
In other words, was it just blind luck that each gamechanger opened the PPF towards accessing the next PPF-expanding gamechanger, or is this reinforcing nature a fixture of human progress? In other words, is it possible that we'll reach an area where the next gamechanger is forever out of reach of our most advanced PPF, say in the way Fusion power always seems to be 50 years away? This could be the case if some technological insight requires, say, an IQ beyond the reach of humans, or energy density levels unattainable with the most advanced technology.
Our history so far seems to indicate that if there are diminishing returns, we are far, far from reaching them. The key concept here is the visibility of new technologies. For instance, consider the brilliant leaps of insight achievable by genius: Leonardo Da Vinci could use concepts extant in his world to imagine flying machines and tanks, concepts centuries beyond his society's energy-wise capability to implement. Despite that, these were visible to Da Vinci.
By contrast, no matter how brilliant Da Vinci was, he literally had no chance of inventing quantum chromodynamics, because the prerequisite nearby intellectual assets were outside of his PPF. It was only through the employment of more energy-intensive means of reality manipulation and measurement that the discovery of the subtle aspects of reality embodied in QCD literally became conceivable. QCD was invisible to Da Vinci. This finally puts us in a position where we can answer your question.
New Technological Progress and Vantage Points on the Path to Kardashev level I
Reaching Kardashev Type I (10¹⁶W) is a hugely ambitious goal, since it would mean boosting our current energy resources by a factor of about 10,000. This is already so far outside our PPF that it is hard to even imagine. How would the world look like if, ceteris paribus, instead of an average US power consumption of 10.0 kW (which is essentially at the current frontier), we would average 100.0 MW? Essentially, you'd have the power of a Nimitz-class aircraft carrier in the hands of Joe Everybody, and that's not taking into account future efficiency gains, which are likely to be considerable as well. Think about it, the power that the most powerful nation on the planet wields (there's only a dozen or so of these behemoths in the world), would be at the fingertips of your average Joe.
This itself would be a greater leap than we've made in all history so far: from a basal metabolic rate of 100W, we got to 10,000 W at the frontier, most of it used to power industrial machines and robotic helpers. In reality, ceteris paribus would not hold, of course, so for instance human population will likely increase greatly before we reach Kardashev I, so the the per capita amount will likely be lower when we reach that threshold, although inequalities in distribution might mean that individuals might have access to levels far above that sooner.
Currently, there is an exploding PPF dimension, where expansion has not slowed for decades:
The implications of this relentless progress in the most scarce resource of all, namely intelligence, are surely vast yet their true scale and the new vantage points sure to be opened by them remain unfathomable. Even a marginally human-like special-purpose artificial intellect would change everything, probably more so than all the previous revolutions combined.
So, even on the path to K-I, there are gamechanging future technologies that we have visibility of, such as fusion power, competent dedicated as well as general purpose AIs, and full-blown nanotechnology. All three are massive game-changers in the energy, knowledge and manufacturing dimensions, which might help explain their relatively easy visibility from our vantage. There are sure to be countless others that remain invisible, some among them gamechangers.
None of these technologies are literally required for attaining Kardashev-I (as the OP mentions, in theory we could reach it with something much like current technology), but it is extremely unlikely that with massive boosts in available energy and the new vantages that opens such innovations will not be pursued, as they will appear as low-hanging fruit in that new context. Having been pursued, they will open a virtuous cycle of making new gains in energy harvesting possible, better computation etc., in turn making yet other potential technologies appear as low hanging fruit, and so ad infinitum or until the marginal rate of return becomes 0.