Staging and launch profile

From multiple sources, comes the suggestion of running 1st stage at TWR = 1.5..1.8, and only use it to push the thing through the dense layers of the atmosphere. Consequentially, those design concepts struggle with 2-stage on moon missions, and have to fly 3-stage for any heavy lifting or interplanetary mission.
Seems there's a much better solution which allows heavy lifting and interplanetary on cheapo 2-stage rockets, and this is it:
--- TLDR ---
1st stage Phoenix max pressure, max throat, min size for TWR = ~2.5, Ramp Down grain, fuel for ~70 s burn time;
2nd stage Pixie max pressure, max throat, max length, min size for TWR = 0.5;
With proper launch profile - circular orbit 100..200 km with remaining dV of 2000 (heavy lifting)..4000 (interplanetary probe).
--- TLDR END ---
Why / how exactly it works.
Few general launch profile problems:
- when lifting vertical, 1G of the whole thrust is wasted into fighting gravity;
- hence, higher TWR results in higher efficiency;
- and the sooner the thing gains orbital speed so the centrifugal force offsets gravity, the less energy is wasted;
- however, can't go too fast in dense atmosphere as it will burn.
So:
1st stage TWR must be much higher than commonly suggested. Efficiency-wise, at TWR = 1.5, only 33% of the thrust is used for actual acceleration, and 67% just fights gravity. For TWR = 2.5 same proportion is 60% / 40% already, and as acceleration is cumulative, and traveled distance is proportional to acceleration squared, the difference in lifting performance is huge.
But once ~3 Mach has been reached, the acceleration needs to be mediated so the speed does not result in burning. On liquid fuel engines it's trivial by throttling, but on solid ones the only answer is a Ramp Down grain (and if that thrust declination is not enough, keeping more vertical profile in the beginning may help limiting speed).
Above ~50..55 km speed limitation pretty much removes, and the pitch should already be close to horizontal, so the majority of acceleration contributes toward orbital speed. Apoapsis at this point should be above 80km, and most thrust goes into lifting the periapsis.
By the end of 1st stage, apoapsis 100..200 km, orbital speed ~1700 (heavy lifting) ..~3000 m/s (interplanetary). Then, to close circular orbit, the 2nd stage should be powerful enough to reach the speed within 2..4 min window, hence the TWR requirement of ~0.5. With higher 1st stage dV, 2nd engine may be scaled down, but efficiency benefit is negligible, while precision and convenience of shorter maneuvering burns is sensible.
With all that applied:
- Interplanetary Probe - $520K + $18K launch (JLP)
- 300kg Lifter +20 Cubesats - $567K + $17K launch (JLP)
- 5t Lifter +Telescope - $4.6M + $2.1M launch (AP)
- 20t Lifter +GNSS - $12.6M + $6.2M launch (AP)
- high efficiency Interplanetary Probe (LF booster) and extreme launch profile - $2.2M + $17K launch (JLP), 6+ km/s usable dV from LCO