This is exactly what miners are doing when checking against double spending!
The efforts of mining aren't to check against double spending, particularly since most double spending attempts are unlikely to ever be relayed to a miner, they are far more likely to be rejected by non-mining nodes before they ever arrive. While miners are nodes too, and miners also perform the same check, the bulk of miners' efforts - the efforts that create gigajoules of heat good for drying strawberries - is to create a record that allows others to define and reject double spending. The verification step doesn't create enough heat to even make a raisin.
Compare to a county recorder's office. The job of a county recorder's office isn't to prevent real estate fraud, it's to record things in a journal. Of course, that journal can be used to avoid fraud, but the job of the recorder is to record.
You have highlighted part of the Wikipedia article, but seem to have underrepresented what it says after the highlighted part: making sure the balances end at the end of a particular accounting period. That has absolutely nothing to do with any of the things that what we call "miners" do. The SHA256 algorithm doesn't help "balance" accounts - it helps create a strong permanent record.
All nodes do this, whether or not they are "mining". If all nodes verify, then what do miners do?