Yeah there's two issues in the article: increased transport costs and delays are going to happen pretty much regardless, assuming the U.K. doesn't stay in the common market.
Technically you're right that it's a choice to impose tarrifs. But don't underestimate how politically impossible it is for the British government to screw over farmers on that scale. If they put British dairy farmers out of business, they become entirely reliant on imports, with almost no control. Agriculture in the U.K. is on a knife edge as it is- it's not about British farmers losing profits but basic survival. When sterling plummeted after the vote a significant number of Irish mushroom farmers went out of business within a few months: they were export-driven, already at v tight margins due to U.K. supermarkets and couldn't lose 20% from anywhere. British farmers are in a v similar position, reliant on the CAP and v exposed to external shocks.
If the gov doesn't increase tarrifs on dairy, presumably the result will be: 1) large numbers of bankrupt dairy farmers, strikes, huge political issues, and massive food shortages. Even if you throw your arms open to the world, the world produces what it can consume/sell and there would be a lag in increasing British supply. 2) massive food instability due to reliance on imports 3) significantly increased prices due to increased transport costs and scarcity- if say 20% of British dairy farmers go out of business, and you therefore lose more than 20% of your supply, you don't have to add in much in extra costs before people start feeling it and politically it's a nightmare- increasing the price of milk, butter, cheese? These are all seen as basic staples.
Again- it's not that there aren't ways around all this. But there aren't any obvious ways around all this that can be negotiated in three to six months that aren't essentially staying in the common market.
Also, there are emergency measures the government can impose under WTO rules if they're in a state of emergency and facing food shortages, which to be frank is likely to be the situation in a crashed out brexit scenario. I think a lot of the discussion on this must be v frightening for people. The issue is, realistically all that will take time to put into place and it's likely there will be a fair bit of disruption first. Not end of days disruption, but certainly the equivalent of a big snow/recession/etc, even if it's short lived.