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It's a hybrid system, THAAD fire control system and radar with PAC-3 MSE interceptor launchers. In this setup the Patriot radar is not involved for longer range engagements. The TPY-2 is really all about increasing range, PATRIOT is very capable of intercepting ballistic missiles on it's own, but is sensor limited when used with PAC-3 MSE. When you want to engage longer range ballistic missiles, MRBM class, at longer ranges themselves to increase defended area and probability of kill, you also have to engage at higher altitude and the PAC-3 MSE has the kinetic energy, but the AN/APQ-65 can't get a good enough track apparently. Which is why PATRIOT is getting a new radar, the LTMADS, so this hybrid system of THAAD and PATRIOT won't be needed in future to max out the performance of PAC-3 MSE. This hybrid THAAD - PATRIOT setup is especially relevant given ammunition stocks of THAAD Talon vs PAC-3 MSE. There are a lot more PAC-3 MSE interceptors in existence than THAAD Talon.

Video showcasing test of hybrid PATRIOT - THAAD system

Can not make stealth ballistic missile

Infrared signature will always be very high and satellite detection of a launch unavoidable in this respect. But the radar cross section of re-entry vehicles is inherently small. Most are conical or biconical and not that large, with small moving aerodynamic surfaces, or no aerodynamic surfaces at all if unguided or using just altitude control rocket motors for course correction. (Israeli Blue Sparrow/Golden Horizon is the latter, Maneuverable Re-entry Vehicle that only uses altitude control system of small rocket motors/thrusters). If the adversary trying to intercept the ballistic missiles lacks early warning satellites, and the re entry vehicle detaches from the booster early enough, it's possible that it could be missed by radar. Look up the requirements for the S-300V/SA-12 air defence system in detecting low RCS objects on re-entry, the Soviets expected Pershing re-entry vehicles to have a small radar cross section (0.05m² RCS).

Video on S-300V air defence system, timestamped interview with Veniamin Pavlovich Efremov, Chief Designer of Krug, Osa, S-300V and Tor air defence systems. States that S-300V was expected to intercept re-entry vehicles with an RCS of 0.05m² at altitudes up to 30-35km

From the book Lightning Bolts - first maneuvering re entry vehicles by William Yengst, a comparison between the RCS of a Tomahawk cruise missile and Pershing-II MaRV.

Our mid-January 1978 meeting with the Defense Intelligence Agency (DIA) was less helpful than Mike, Jack or I had hoped. Two subjects were on the agenda: (i) in-flight survivability of Pershing-II versus Tomahawk (GLCM) and (2) distribution of important targets in Warsaw Pact nations. The meeting began with a blunt declaration by a DIA senior analyst, who claimed, "Warsaw Pact radar systems would never see Tomahawk cruise missiles."He declared the radar cross section of the cruise missile, as seen from the forward hemisphere, was under 0.1 square meter (roughly one square foot), similar to the size of a "sparrow." With no measured data but only analytic estimates, we contended the cross-section of Tomahawk was closer to a few square feet nose on, increased near the beam to over 10 square meters (i.e. about 100 square feet) and reduced somewhat through the rear hemisphere. The wings and tail surfaces would have a large radar cross section, particularly if the missile flexed or vibrated during flight. By contrast, experience with observable signatures of reentry vehicles suggested Pershing-II would have an order of magnitude smaller radar cross section throughout its reentry trajectory.

but mach ~10 will begin to weaken signal by plasma mach ~15 will produce fₚₑ > radar frequency and block but different problem and no radar solution)

The Lightning Bolts - first maneuvering re entry vehicles book has some interesting stuff on tracking objects moving at high hypersonic speeds through plasma, I forgot what exactly but it's something the US was very interested in during the cold war.