The authors explicitly suggest the first part of your question in their discussion section:
indicating that full–length spike–based vaccines may inhibit the recombination of V(D)J in B cells,
However, for this to have a meaningful effect on the immune system as a whole requires that a significant portion of B cells uptake the vaccine and express spike protein, which isn't demonstrated. The paper also, frustratingly, doesn't indicate if their results showing an effect of spike protein on DNA-repair occur in the presence of the non-structural proteins Nsp1, 5, etc., which as they mentioned are known to transport themselves to the nucleus. If spike protein can't get to the nucleus without these other viral proteins which are not included in the vaccine, then there's no reason to suspect the vaccine-produced spikes would have this effect.
With regards to the second part, "rendering our immune system more vulnerable in general than it was before", the spike-based vaccines have been demonstrated to generate a robust B-cell response, so we can rule that out. The immune functions full-length spike protein is suggested to impair are measurably functioning well in vaccinated individuals. In the cytoplasm of cells, mRNA of vaccines and the spike protein they produce can be expected to be recycled after a day or two, so long term effects of this nature are very unlikely.