I think that the cost of hiring and training election workers is a major issue. I'm not sure how European countries handle this, but in my jurisdiction election workers are hired as temps, paid hourly, and receive a several hour training. Managing this is a formidable task for the fairly small election bureau, which has a limited budget because, well, elections only happen so often.
Just the staff that processes absentee ballots (on normal years, even more so this year) is a large group of people who operate 24/7 in a large warehouse. We're fortunate that the county has the resources to dedicate a large building to the election bureau, in other counties they have to borrow space in community centers or courthouses for counting and it's more of a logistical challenge to get equipment in/out. Yes, many of these same challenges exist and are managed for the physical polling locations, but hand-counting would be piling a lot more workforce and space to manage (I would guess about double) on a county that is already struggling to afford the operation. The use of precinct tabulators keeps the polling-place staff fairly small while meaning that the ballots are already tabulated when they are delivered to the election bureau.
I have been involved in hand-counting ballots and it is remarkable 1) how tedious and time consuming it is, and 2) how error-prone the process is. With our typical ballots (18" long paper with three columns of questions on both sides, typical Dominion design), it takes a while for each ballot and it's a two-person process for accuracy. Even then the election administration feels that the error rate is unacceptable and so takes every measure possible to avoid ballots that must be hand-tallied. Much of this is a result of election code requirements for accuracy that involve typically three-person (and two-party) sign-off on every tally operation. Security restrictions on access to the computer totaling system adds further complexity and has lead the county to abandon "distributed" hand-tally operations, with all handling now being done centrally in the warehouse.
My gut feeling is that countries which hand-count ballots probably have a lower emphasis on absolute accuracy than US election administrators typically do. I have no idea but would also be interested in knowing how common it is for election workers to be paid in Europe vs in the US. My gut feeling, though, is that hand-tallying is not practical in the US without a significant expansion of the budget for election administrators. I'm also unconvinced that hand-tallying has any meaningful advantages over precinct tabulation, since both allow for post-tally audit and in this jurisdiction and many others, hand-tally audit of a sample of machine tabulation is a standard part of post-election canvassing.
As an addendum on something somewhat poorly understood, in most US jurisdictions the long delay between "unofficial" and "official" election results is a result of the canvassing process, which is a combination of hand-processing "difficult" ballots like provisional and mismarked ballots, and a standardized post-election audit which may include hand-tallying a sample of the machine-tabulated ballots (I would like this to be a mandatory practice nationally). Much of this wait is not because the machine tabulation is slow, but because the machine tabulation is not completely trusted and there is an auditing process that takes some time, plus there being ballots which are inherently not suited for machine tabulation. In my area the unofficial tabulation is usually completed overnight after election day, but the canvassing process takes substantially longer because, among other things, every provisional ballot must be manually researched by a more highly trained records team, which is a measure to avoid denying anyone the right to vote because of procedural problems with looking them up in the pollbook.
Finally, as discussed, not all states allow absentee/mail ballots to be processed before polls close. There's a tradeoff here and I don't think this rule is entirely unreasonable, but it does obviously add to the delay.