There are two strategies at play. The first is PR/propaganda playing to a US audience, midterms etc and most importantly keeping Magats on side. Trump needs a win.
The second is Trump trying to find an off ramp. Control of the SoH is Iran's ace, but it also needs to sell its oil to finance war/economy, so taking the SoH will hurt Iran financially at a time when its assets are also frozen. Which might force them to the negotiated peace table etc.
Re countries paying a toll. Apparently this has been done in the currency which must not be named. Totally untraceable. If I were Iran, I would secretly contract with interested parties to not launch a drone or two while their tankers were passing through, while letting the US think they're successfully policing it.
How the US intends to actually do the blockade is a 'watch this space' answer.