In essence, yes, the alpha chain is more important than the beta chain, because there are alternative forms of hemoglobin that can deliver oxygen without beta chains, but the hemoglobin forms without alpha chains cannot.
Hemoglobin has a bunch of different forms, depending on which four globin chains make up the protein. While there are even more, here's an overview of the most important forms:
* Hemoglobin A is α2β2, this is the normal form, and in adults this makes up 95% of our hemoglobin.
* Hemoglobin A2 is α2δ2, with 2 delta chains instead of beta chains. It normally makes up about 2-3% of hemoglobin, and its properties are nearly identical to HbA.
* Hemoglobin F is α2γ2, with 2 gamma chains instead of beta chains. This is the primary form in utero, because it has a higher affinity for oxygen, and thus can extract oxygen from the maternal circulation. It normally decreases after birth, and makes up <1% of adult hemoglobin.
* Hemoglobin H (β4) and Hemoglobin Bart's (γ4) are defective forms of hemoglobin that occur without alpha chains. 860
Both of these are unstable molecules, and are poor at delivering oxygen because their affinity for oxygen is too high (not allowing it to be released in the tissues).
What you'll notice from the above is that there are functional hemoglobin variants that don't have beta chains, but no functional variants without alpha chains.
If someone has beta thalassemia major, with loss of both β alleles, the body compensates by ramping up production of HbA2 (with delta chains) and HbF (with gamma chains), which compensate for not having HbA. That's not to say that everything is okay... it's still a very serious condition, and usually progresses to death before age 20 if untreated. But, the hemoglobin variants are sufficient enough to allow people to live.
If someone loses all 4 alpha alleles, the only hemoglobin they produce is Hb Bart's, which isn't functional. There are no alternatives to the alpha chain like there are with beta thalassemia, so there's no mechanism to compensate for this loss. Because forming HbF isn't possible without alpha alleles, this condition is fatal in utero. 860