Okay, you're conflating the customs union and the single market.
In simple terms the customs union means that we would apply the same external tariffs to non-EU countries as the EU and would have zero tariffs between us and the EU. That simplifies trade with the EU but would prevent us from striking trade deals with other countries based on reduced tariffs. Because if we strike a deal with the USA on whiskey whereby we charge zero tariffs on bourbon and the EU has a 25% tariff on bourbon then the EU would have to check UK freight to make sure that no one is channeling the cheaper bourbon through the UK into the EU. That's an example of one product, obviously there would be 1000's involved in any trade deal.
The single market is what harmonises regulations throughout the EU so anything made by one country is automatically acceptable anywhere across the EU. Chequers proposes that while we aren't in the SM, we would copy the SM rulebook on goods. We could potentially strike trade deals based only on regulatory divergence from the EU standard but a. this isn't a massive advantage for us, tariffs are much more of an issue and b. it would means accepting products that are below the standards we currently have i.e. it's the consumer who loses out.
Your idea that we simply don't check anything coming from the EU isn't feasible in practice. Firstly, it would mean that under most favoured nation rules we would be unable to check imports coming from any country in the world. Abandoning checks for the EU means abandoning checks for everyone. That would most likely lead to a lot of dodgy stuff coming into the country as well as no checks on whether the correct tariffs have been paid. Secondly it doesn't solve the issue of EU checks on freight entering their jurisdiction. On 30th March a lorry coming from France may get into the UK with no delays but it won't get back out without massive delays. This then has a knock on effect on their next journey into the UK not to mention the fact that fresh produce from the UK could be rotting by the time it gets into the EU. SO JIT industry that relies on the UK in any way fails. It simply doesn't work unless both sides drop all checks, and the EU isn't going to do that for anyone outside of their CU and SM.