Is the "cosmological constant tension" the prime reason that we believe the expansion of the universe is accelerating?
No. Evidence for the accelerating expansion of the universe comes from multiple angles: supernovae data, Baryon acoustic oscillations, the mass functions of galaxy clusters, etc. That the universe's expansion is accelerating is not in doubt; the question is by how much.
There is a potential tension between low redshift probes of mass clustering and Planck data (CMB measurements). This ongoing speculation might be evidence of new physics or even modifications of general relativity. However, the author of the article you cited seems to have a confusion between the cosmological constant (no tension discussed in the literature) and the Hubble constant (that has $3\sigma$ differences), which becomes clear when you read the comments where a noted CMB researcher expresses his views. There is even no real evidence that the cosmological constant is not constant.
So it is really exciting if some tensions in CMB data grows into significant differences. But one should not confuse the cosmological constant with the Hubble constant which are completely different stuffs.