We compare seven internal-linear-combination (ILC) and scattering-transform (ST) estimators for thermal Sunyaev-Zel'dovich (tSZ) recovery on FLAMINGO lensed simulations with explicit Simons Observatory and Planck noise, and report a unified pipeline that wins on three axes simultaneously. The seven methods are: harmonic ILC, FoCUS, and STsep at the per-patch effective beam; pixel ILC, needlet ILC (NILC), and constrained NILC (cNILC) at a common beam; and STsep, a pure multi-channel ScatCov estimator with SED-difference initialisation that uses no ILC weight at any stage. On 20 patches at 150\,GHz, cNILC wins pixel correlation (, over pixel ILC); STsep wins relative RMS and KS distance (, KS ); FoCUS sits within patch-to-patch scatter of ILC and is a refinement, not a paradigm change. We introduce a third-axis diagnostic, the recovered cluster-centre amplitude on the deepest tSZ pixel of each patch: cNILC , ILC variants --, STsep , STsep only . Three ablations (amplitude-prior, optimisation length, peak-aware loss) all falsify candidate explanations and isolate the as the intrinsic ScatCov optimum under SO+Planck noise rather than an algorithmic deficit. The companion band-pass plus Cholesky post-processing (BPCholesky) brings every linear method to RMS -- and calibrates cluster-centre amplitude: post-BP, six linear methods reach -- of truth (cNILC ). A new ST-guided posterior refinement of cNILC adds a ScatCov-class correction; stacking BP and ST-refine on cNILC gives a unified pipeline cNILCBPSTHM that simultaneously reaches cluster-centre amplitude and a ScatCov-distance reduction relative to raw cNILC at \,s additional GPU cost per patch. The deployable analogue uses ensemble-mode BP with a peak-clip post-process (no per-patch truth at inference) and reaches central recovery. We propose the unified pipeline as the practical default tSZ compsep recipe when both cluster amplitude and non-Gaussian morphology matter; STsep remains the right pure-ST demonstration but is amplitude-biased for cluster cosmology. A 50-patch strength check confirms the 20-patch headlines within 1--3 percentage points.