Yes, the ones YOU built, maybe. But what about the links (sometimes thousands of links) of the links built to your site what you didn't build?
I've been doing this a long time, and I can tell you that GSC only shows you a percentage of the links they know about.
Whether I built the link or someone else did doesn't matter. Google doesn't discriminate. Nor could they even if they wanted to since they can't tell who built what link.
The reason I even mentioned that I built the links is because I would be looking for those links. And in those cases, 100% of the time I've found my links. Which is my verified sample. I can't verify the other backlinks because I didn't built them and therefore, have no way to know if they exist or not. So that sample is not verifiable or accessible to me.
Now, am I correct in assuming that GSC shows 100% of the links they know about based on my sample? It is very possible that I am not because my sample size is small. But there's no evidence presented in this thread yet that is the case.
I value the truth over being correct. So if you have evidence, I'd love to see it so I can adjust my knowledge and understanding.
Also, just so people understand, I don't check "Download this table" or "Download more sample links" in GSC. Those will not be exhaustive. GSC says as much. What I look at is "Download latest links" and I have a running total month after month. So far, that has been exhaustive and encompassed my sample size of links.
However, if you are making the point that he can't see his full history of backlinks from just GSC, now that I think about it, that must be true as his "Download latest links" will only be exhaustive for, what I would guess to be, 2-3 months at most. So yes, he should probably use Majestic or Ahrefs for historical links. That's a good point. GSC will be more exhaustive though between those three tools in my experience. I also only have experience really with Majestic by the way as I chose that service over ahrefs when I tested them out around 3 years ago.