Double patterning does not require double exposure... Only if you need to rework, and that is the same for any litho step whether pattern multiplication processes are used or not.
The higher cost comes from the subsequent processing needed to reduce the line feature size (Etch/trim, oxide dep, etch/resist clean, and final etch/dep)
The increased processing adds additional complexity (Defects and margin for error). But does not increase Litho tool utilization or need for additional exposure. The design is a bit trickier in terms of routing, but vias and inventive design can solve that.
As for TSMC strategy whether disputed now or not seems to be the course they are on. A supportive point to their strategy is everyone(Memory makers, Samsung, TSMC, Global, Intel, etc) was able to progress with DUV past 39nm(limit of DUV) with these tricks for nearly a decade, When TSMC is at the wall of what EUV can accomplish without patterning tricks, there should still be plenty of mileage to get them another 4-5 years, and it might be the cost optimized route with how out of hand Litho BOM has became in Semiconductor Fabrication. Although it may give Intel the lead again in node advancement(Intel should have an easier way of achieving smaller features and do so faster), but at what price?