Lightning Network, as it's described, seems to solve all the problems with Bitcoin.
That's not true. I would even say that Ligntning Network doesn't on its own solve any of Bitcoin's problems, but it does go a long way towards solving several of them (scalability and privacy).
I'm unable to accept payments due to the insane fees, but would be able to if Lightning Network were supported.
There are invoicing solutions that support Lightning Network, for example BTCPay Server. Considering Bitcoin Core as the only "serious" Bitcoin-related software is short-sighted as there is absolutely no reason for that. For example, I doubt many people would prefer storing their life savings in Bitcoin Core instead of a hardware wallet.
Why is not the "real" Bitcoin project working on its own "layer 2", as a core feature of Bitcoin ... ?
This is where I cover the rest of the question.
Bitcoin Core is the most widely used implementation of a Bitcoin node, but it's not the only one and anyone is free to implement their own version, as long as it follows the same protocol rules.
Similarly, there are multiple implementations of a Lightning node, and perhaps over time we will see the ecosystem converge on one of them being by far the most widely used, however there is no reason why it should be developed under the same project as Bitcoin Core.
Quite the opposite, I don't believe Bitcoin Core developers should leverage their position in order to favor any specific Layer 2 solution over another, and implementing their own into Bitcoin Core would be doing exactly that. I think it's much better for Bitcoin Core developers to bring upgrades that all Layer 2 solutions can take advantage of, such as Taproot.