Everything is full of humans, but private companies with terrible processes workers would usually have consequences that demand some change. Government services have no competition sos they are exempt from consequences if they're terrible.
Before the war when I studied Russian, I had the unpleasant experience of having to get a Russian visa, which is usually subcontracted to "VFS Global", a private company. Didn't do anything to improve the experience at all.
Plenty of exceptions to that – broadly speaking, oligopolies, but generally anywhere with low competition like the only convenience store in a locality, rural broadband, etc.
There is no perfect competition, supply doesn't perfectly meet demand, and incentives aren't always perfectly aligned ¯\_(ツ)_/¯
In any case, government services change through policy and, thus, indirectly, through the electorate. But a) it lags, b) is not the only issue the electorate care about, c) the supply of money is not infinite, d) the supply of labour is not fungible and e) in this specific case, policy on immigration is generally driven by people not immigrating, so the incentives are not aligned.
Can you read my previous point? Vodafone Germany has competition, I'm not forced to be their customer if I don't want to, but I can take my money elsewhere.
Meanwhile I can't choose between other government services while in Germany but those terrible forced upon me by the state.