lazy with their monitoring and didn't notice someone walking out with everything. And lazy about separating prod and dev environments. as someone who breaks in to network for a living, i can assure you this is not unique to lastpass lol.
as far as i know, lastpass breach was decently done. the dev was running a plex server at home. someone threw a 0day RCE at it from the outside and used that to pivot into his network. i dont think they ever disclosed how the attacker went from network access to actually owning the dev's box. but from there, they waited until he already logged in to the vpn (after he put in his MFA), and rode the connection in that way back to the lastpass network.
so far so good. lastpass did everything right and the attack took a lot of resources to carry out.
but where they fucked up was the dev having access to the prod data. i'm guessing they did the right thing and had all the data in multiple accounts (separate AWS account for prod, different one for dev, different one for staging). no access to each other. cool. but the dev probably had access to both (or all?) of the accounts, so the attackers could just ride in to the prod account as the developer. again, i haven't seen any details on this part of the breach, but i've seen (and done) this exact thing (well... similar -- SE to get code execution on developer company workstations while they work from home - not attacking personal devices. that's illegal) many times.
you absolutely could physically write down a long password on paper for person use at home (as long as you can keep it secure). you are infinitely more likely to get your box randomly hacked than a fucking thief breaking in to your house, finding your password paper, and getting away with it. if they did that shit, you have way more problems to worry about than losing passwords.
this is basically a discussion in threat modeling. who is mostly likely to attack you? how will they do it? is writing a password on paper perfect or great? no. is it less likely to be stolen than a file on your computer? yes... or probably. as long as you have a safe place for it that your wife/kids/dog can't get to it or damage it.