Bank treating loan as asset

A loan is an asset, not just to a bank but to any business. It is the same as an account receivable in that it anticipates the value of future cash flow from repayment. Thus it holds implied value (no differently than cash, since currency only has value so long as its issuer can make good on their obligations, thereby creating a trust relationship between the issuer and the party willing to accept it as a medium of exchange). It only becomes a liability (or a "non-performing asset") when the borrower fails to make the payments.

In the sense that a loan impairs a bank's capital reserve, you would be right not to see it as an asset.

Before lending some of your money out you have the money. That's an asset. After you lend the money out, you no longer have the money but you have a different asset: the fact that someone has promised to repay you the money and pay you interest. Exactly what that asset is worth is, well, complicated. It's worth more than cash because it will get you interest. But then again it's worth less than cash because there's a chance the borrower will default. This is why banks charge higher interest rates to riskier customers: they want the "higher" and "lower" to balance out so that overall, a loan of 10K is worth slightly more than just having 10K on hand.

Calling the loan an asset even though it involved money leaving your reserves is no different than calling a car or a work of art an asset. Money left your hands to acquire it, but acquire it you have and it is worth something to you. That's an asset.