https://theconversation.com/covid-vaccines-focus-on-the-spik... is probably relevant to your question. The short answer (this was also discussed recently on TWiV) is that spike is the earliest interaction with cells, so focusing on it hopefully causes the immune response to mount earlier as well. But, as described in the link, other proteins may be viable targets as well.
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Another issue is, other targets may not provide neutralizing immunity and may facilitate ADE. I don't think we've seen ADE in natural infection yet, so it may just be a theoretical issue, but I also don't know if we've really looked for ADE in natural infection either.
We do know that other similar coronavirus vaccine candidates (a candidate for SARS-COV-1) exhibited ADE in primate studies. So since the spike protein appears effective currently, maybe it's best to stick with that.
For people curious about ADE (a fascinating topic that so far doesn't seem to have caused problems with SARS-CoV-2), here is some excellent background reading that covers many of the points made in the above comment, with citations and deeper discussion:
https://blogs.sciencemag.org/pipeline/archives/2021/02/12/an...
I remember when this whole thing originally broke out, ADE was my biggest fear. Even when China was talking about a 2-3% infection fatality rate, it's like ok.. well that's not going to shatter society. But ADE... the prospect that getting it a 2nd time would raise the IFR to who knows... 20+%? That was scary.