One thing missing from this thread is how building laundry card economics interact with replacement cards, management turnover, and whether the balance is stored on the card or in an online account. I've lived in four buildings with card-based laundry, and the only one where bulk loading made sense was a building with an online reload portal where balances were tied to my unit number, not a physical card. If your card is just a plastic card with a magstripe and the balance is stored on the card itself, treat it like cash: don't load more than you can lose or have frozen in the machine during a holiday weekend when the super is off.
That said, the per-load savings are easy to calculate. If the bonus is $2 for every $20 and a wash plus dry is around $5, then $20 gives you four cycles plus a $2 bonus. In many systems that bonus only applies as credit, not cash back, so you're really getting part of a fifth cycle after you spend $20. Over a year, if you do 100 cycles, that could be meaningful, but only if you actually do all 100 in the building. If you move, or the machines are down for two weeks, that's lost.
Since you're new to the building, use $10 or $20 loads for the first few months. Track how often you actually use the dryer, especially in summer when a drying rack works fine. Also check whether the machines charge extra for high heat or heavy cycles. The 'bonus' can disappear if you have to run something twice.
For your work shirts, be careful with wash-and-fold places. Many use industrial detergent and high heat unless you pay extra. Cheap tees are fine, but structured shirts do better on cold in your own machine and air-dried. Woodhaven_W is right about the house card too—supers often have one for testing machines, but don't rely on it.
To answer your original question: for two loads a week, start with $10 or $20, not $30. You're not crazy to hesitate. The discount is not enough to justify handing over a month of laundry money upfront in a building with unknown machines and a card you've already lost once.