  The Fred Young Submillimeter Telescope (FYST) and the Simons Observatory
Large Aperture Telescope (SO\ LAT) will deliver unprecedented high-resolution
measurements of microwave sky emissions. Notably, one of those microwave sky
emissions, the thermal Sunyaev-Zeldovich (tSZ) signal, is an essential probe
for cluster astrophysics and cosmology. However, an obstacle to its measurement
is contamination by the cosmic infrared background (CIB), especially at high
frequencies. Our goal is to assess the detection and purity of tSZ power
spectrum measurements from these two telescopes. We demonstrate that FYST's
high-frequency coverage helps lower CIB contamination and improves signal
detection. We simulated the various components of the microwave sky at the
frequencies, sensitivities, and beam sizes of the upcoming SO LAT and FYST
telescopes using full-sky Hierarchical Equal Area isoLatitude Pixelisation
(HEALPix) map templates from the Websky simulations and the Python Sky Model
(PySM). We used a map-based internal linear combination (ILC) and a constrained
ILC (CILC) to extract the tSZ signal and compute residual noises to assess CIB
contamination and signal recovery. We find that the CIB's residual noise power
spectrum in the ILC-recovered tSZ is lowered by $\sim 35\%$ on average over the
scales $\ell \in [500,5000]$ when SO LAT and FYST are combined compared to when
SO LAT is used alone. We find that when using CILC to deproject CIB, the
combined abilities of SO LAT and FYST offer a large $\ell \in [1800,3500]$
window in which the recovered tSZ power spectrum is not noise dominated.
