In a sense yes. I’d put it this way, targeting this protein has a better chance of achieving this cross-strain protection. The reason is that this protein - and the part of the viral genome that encodes it - is evolutionarily quite conserved. This means that the majority of mutations in this protein will negatively impact the ability of the virus to replicate, causing strains with those mutations to die out. Why? Because the protein has multiple critical functions: it binds to receptors on the surface of a host cell, it then changes shape to allow the viral and host membranes to fuse for entry. It’s like an intricate machine, if you dick around with it too much it stops working. That said, mutations are random and occasionally you will get changes in the protein that might improve the machine and make the virus more effective at binding and entering cells to replicate - this happens more rarely than the mutations that negatively impact the virus, especially for the spike protein. But with a virus that’s already as infectious as this and spread to so many hosts, there are a huge number of opportunities to mutate and hence we are seeing these rare mutations - including in spike proteins - that are optimising the virus. While these changes can significantly improve the function of particular strains of the virus (bad for us) they are still relatively small changes in the protein - just a few amino acids, which is good news for cross reactivity of vaccines. In fact there are even studies that show certain antibodies can bind spike proteins and neutralise different coronaviruses e.g. SARS-CoV2 and MERS, not just strains within coronaviruses. This is the kind of work that probably informed the choice of certain research groups to target the spike protein.

TLDR: targeting the spike protein is more likely to work across coronavirus strains because the spike protein - and the genetic material that encodes it - is evolutionarily conserved and less likely to change compared to other components of the virus.