Weight and balance calculations are something that anyone in aviation will wind up memorizing early in their career. You don't "need" bespoke software for it. You don't need software at all. It just makes it way easier when you have hundreds of individual items.

What's most likely to happen is that they will punch it all into an excel sheet and get their primary objective of weight distribution correct. What will suffer is cargo organization so the crew at the layover is likely to find that they need to unload a bunch of stuff in order to get at some cargo that needs to come off at that stop.

A plane "could" fall out of the sky but that would require multiple people not doing their job but there's no point in pearl clutching over that because even if the software works the same people not doing their job can still cause that outcome.