Can somebody explain to me if Taproot will fix the issues that Lightning Network solves, without the Lightning Network?
That is, near-zero fees, privacy and fast settlements?
Will Taproot allow me to skip Lightning Network and simply use pure Bitcoin Core?
Will Taproot allow me to skip Lightning Network and simply use pure Bitcoin Core?
Well, Bitcoin Core doesn't support Lightning for now, and probably won't for the foreseeable future, but there is no reason why that can't change over time. I'm going to assume your question is:
Will Taproot allow getting all the benefits the Lightning Network provides with just on-chain Bitcoin transactions?
Absolutely not.
Taproot is an improvement to the script infrastructure that exists on chain. It makes some transactions slightly cheaper, in particular more complex ones, and most importantly makes many types of transactions look indistinguishable, improving privacy.
However, on-chain transactions still suffer from all the same problems that make it much more appropriate as a settlement system for higher layers than as payment system directly:
These are side effects from the fact that on-chain transactions are broadcasted and verified to the whole world, and the whole world needs to get consensus on their occurrence. Lightning solves these by building on the security properties of the on-chain system, and building a fast, reliable, cheap, payment system on top at the cost of additional setup and additional assumptions. You simply can't build a usable payment system where every transaction needs to be broadcast and verified by the whole world - Lightning solves this by making the payments simple participant-to-participant interactions, and only broadcasting settlements to the whole world, or to counter misbehavior.
Lightning however can take advantage of the new features Taproot offers, and be upgraded to use it. In particular, it's possible to make non-adverserial Lightning settlements looks like ordinary payments on-chain.
With respect to the original wording of the question I believe Taproot will actually make the Lightning Network pointfull instead of pointless.
While Taproot allows us to save some space on the bitcoin base layer, it will not make the Lightning Network obsolete. Remember, the main issue with respect to on-chain scaling is that a protocol in which every transaction is broadcasted to every participant is just too inefficient if those transactions should represent a substantial fraction of monetary payments in the world.
With respect to the Lightning Network there are actually many improvements possible for the Lightning Network if Taproot is activated. Most notably, we can have Point Time Lock Contracts (PTLCs) instead of HTLCs which give us way more flexibility in the payment process as the addition of points and secrets is linear whereas the addition of payment hashes and secrets is not linear. E.g.:
H(r1+r2) != H(r1) + H(r2)
The property of linearity allows us to have applications like
