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The whole thing is fucking abysmal, I hate it. It's ceramic so I would probably enjoy lobbing it into the garden and taking my 5lb lump hammer to it. But as you suggest, I will probably swap the tap out first and see if I can sort the sealant around the edges first as that would be the cheaper/easier option. Although, from the initial glance I took the other day, access to the tap nuts looks pretty tricky. Post lump hammer, access should be much easier.
Change the tap first - taller, shorter, longer - anything. You may get a much better sink.
We moved in and out kitchen had the Ikea Belfast sink knock-off. Hated it. Changed the tap and now love the combo.
Taps are easy to change over.