The second part of your question is the easiest to answer, how much manual work is involved in settlement processes?
Payment systems which handle low value (i.e. high volume) transactions work on the basis of net settlement. Each of the individual payments are netted across all of the participant banks, so that only one "real" payment is made by each bank. Some days banks will receive money, others they will pay money. This is arbitrary and depends on whether their outbound payments exceed their inbound payments for that day.
The payment system will notify each Bank how much it owes/will receive for the day. The money is then transferred between all of the banks simultaneously by the payment system to remove the risk that some pay and others don't.
If you're going to make or receive a very large payment, you're going to want to make certain that its correct. This means that if there's a discrepancy, you need operations people available to find out why its wrong.
When dealing with this many payments, answering that question can be hard. Did we miss a payment? Is there a duplicate? Etc.
The vast majority of payments will process without any human involvement, but to make the process work, you always need human brains there to fix problems that occur.
This brings me to your first question. On every day that settlement happens, a bank will receive (or pay) a very large sum of money. As a settlement bank you must settle that money - the guarantee that every bank will pay is one of the main reasons these systems exist.
For settlement to happen, every bank has to agree to participate, and be ready to verify the data on their side and deliver the funds from their account.
So there is no particular reason that this doesn't happen on weekends and holidays other than history. But for any payment system to change, it would require the support of (at least) a majority of participants to pay staff to manage the settlement process on weekends.
This would increase costs for banks, but the benefits would only really be for you and me (if at all). That means it's unlikely to happen unless a government forces the issue.