It doesn't hurt the government, though. The first order effect is new tax revenues they don't have to source elsewhere.

As you pointed out, demand will drop, so there's fewer sales, and it drives inflation, slowing the economy, leading to (ultimately) a reduction in revenue all around. Also, it's basically a heavy sales tax, since it's through purchasing that's it's implemented, and sales taxes are always regressive (the poor spend all their money on consumables, the rich spend only a portion, so the poor pay more proportionate to their income).

There's a reason no economist these days is in favour of tariffs.