Will "Taproot" replace Lightning Network? 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?
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

stuckless payments
payment decorrelation
escrow services