The purpose of minimizing convexity is to match the cash flows with the liability. If we only aim to pay off a single liability, ideally, all of the cash flows should mature on the same day as the liability, which would be an exact cash flow match. The more dispersion (i.e., cash flows, convexity) that are further apart from the liability, the more risk (think reinvestment risk, because this is all based on IRR. More convexity is expensive and opens us up to structural risk). The best way to immunize a single liability is to match the PVA to PVL or match the Macaulay duration. Minimizing convexity is a decision you make if the other 2 are sufficient in meeting the liability.
When it comes to multiple liabilities, we look for portfolios that have convexity/asset dispersion slightly greater than the liabilities. The idea is that we don’t want too much convexity because it then opens us up to structural risk (more disperse cash flows being reinvested at different rates), so we try and match the convexity to the liability convexity so if there is a structural change, we closely match where the liability portfolio ends up.
I hope this helps!