I've been using GNUCash since 2007, while in grad school earning... grad school wages. It worked out well for me earning IT specialist wages now, and I don't think the regular joe will need/use plaintext accounting specifically. But the idea of double-entry bookkeeping can be useful still.
1. Accurate visualization. As you suggest, a great number of people pay most attention to their checking account balance. Making student loan payments on time can feel like just paying 'bills'. But it's actually a split between interest and principal, and in a sense, you are accumulating wealth by paying down principal in a way that just looking at the debit card balance will not. Similarly, racking up CC debt can feel cheap when making minimum payments.
2. Improved forecasting. Before GNUCash I largely worked the same way as you describe. Keeping a mental tally of bills outstanding and checking my balance via the automated phone system to judge what I can spend. Now I have scheduled transactions for those bills recorded 90d in advance, and can see at a glance my Minimum Future Balance. It's nearly the same thing, but with way less acid reflux from stress. And I can schedule in all the random annual expenses -- vehicle registration, vacations (when permitted), annual subscriptions -- just once and the system remembers for me. Plus I have a more granular archive of expenses to estimate and budget around.
3. Expense Analysis. Most people track one asset: checking acct, and maybe liabilities. Double entry accounting makes it feasible to transition from that to treating expenses and income as equally important. Probably the canonical example is car insurance; most insurers offer a six month premium, or a monthly payment with a surcharge. If you focus on checking acct balance, you go for the monthly payment with smaller upfront. If you want to reduce your annual expense you pick the six month option, dipping into savings briefly if required. On the income side, you'd be surprised how many people don't even bother to get the employer 401k match, even though you can often carry a CC balance (a very worst case scenario) and come out ahead.
4. (advanced) Portfolio balancing. If you're fortunate enough to fund a retirement plan, tracking your portfolio and asset allocation can be tricky across Roth IRA, 401k, etc. An annual rebalancing, formalizing the 'sell high, buy low' formula, is equally tricky when none of your accounts synchronize. A double entry system makes this and other retirement planning steps feasible.