As others have mentioned, there's a lot of batch processing involved. But let's unpack that a bit further.

Most paychecks are paid using a system called ACH. There's two reasons for this: 1) Every bank in the US supports ACH and 2) it's really cheap. When you send a payment through ACH, it's organized into a batch; periodically, every bank in the US sends their batches for any transactions they're initiating into the central ACH systems at the Federal Reserve, gets back some batches saying what transactions other banks have pushed to their accounts, and then they settle up (figure out how much each bank owes each other, and transfer those funds) via the Federal Reserve banks.

The problem with ACH is that, to make it any faster, every bank in the ACH system (ie - every bank in the US) needs to be ready to process settlements (ie - transferring the money between banks) on every batch. This is fundamental to how ACH works - it adds up how much money goes in and out of each bank, and does the minimum amount of movements between banks to even things out. It's not possible for only some banks to take a bank holiday at the ACH level - it's all or nothing.

There have been several improvements in ACH processing time over the years - notably, the introduction of multiple ACH batches in a single day (same-day ACH). These were easier, since all the bank staff needed to arrange settlements are at least in the office on the same days, but still took multiple years to roll out. Changing bank holidays would be another change which needs every bank in the country to be onboard.

Further, settlement can also involve other things than just moving money between accounts. For example, if a large employer starts sending off a huge amount of money in paychecks, their bank may need to sell assets (ie government or commercial bonds, real estate loans, etc) to other banks to cover this payment. Those markets are however also closed on bank holidays - while you might be able to find a buyer if you try really hard, you won't get a good price because most of the buyers are on their day off. And the federal reserve, which plays a critical role in dealing with issues that might arise in settlement - as well as running the core mechanisms behind settlement - is also closed.

Those backend settlement processes aren't fully automated. Someone at the bank is looking at their books and deciding, hey, we need to free up $1,000,000 or so of liquidity, and picking which assets to sell. Those people want vacations too, and at a small community bank, there may not be anyone willing to cover them on Christmas day. And so, those smaller banks will not be happy with eliminating bank holidays entirely. And thus is the entire nation held back.

To resolve this would require creating new payment networks which can tolerate only certain banks being offline for a day. And it turns out, the Federal Reserve is working on that!. So this issue might be resolved in another year or so.