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.