The second dumb assumption is that we can source cheaper food from elsewhere by unilaterally dropping tariffs. This ignores the fact that the EU Everything but Arms (EBA) agreement, in force since 2001, allows all imports from the Least Developed Countries duty-free and quota-free, with the exception of armaments.
The reason LDCs struggle to take advantage of it is the non tariff and regulatory barriers. In this, we do not have a free hand in relaxing standards because we are either pegged to the global standard or will need to maintain close harmony with the EU as our nearest trading partner. This is also on the assumption that supermarkets will even by non compliant goods. With any waiver comes the risk of increased inspections and added costs which may nullify any price advantage.
Central to Mrs May's stance is a number of bogus assumptions about how trade works, the leeway we will have once we are out, and the more disturbing reality that other countries might not even want our trade.
In most respects the world has reached a state of trade normalisation where any agricultural nation in the developed world is already running at capacity on established trade lines and may be unwilling to expand capacity without passing on the investment costs directly to us. They won't necessarily be able to supply what we want at the volumes we need.
The idea that we can flounce of and be a "global Britain" completely ignores the fact that most nations now have some kind of cooperation agreement with the EU (or the USA) and are consequently bound by those regulatory obligations - and are not equipped to make exceptions for the UK which now seems determined to be the odd one out.
If we are also dead set on a process of deregulation then that precludes the possibility of grandfathering EU deals in the long term. That then starts the clock on rebuilding our trade and diplomatic capabilities from scratch in order to reacquire what we already had and could have kept had we stayed in the single market. So now we have lumbered ourselves with the mammoth task of administering the Brexit process we are also forced to march at double pace just to stand still.
Before reading the May speech in full I had a feeling that it was an opening ploy, but in fact it is marked by naivety and colossal ignorance to such an extent that it could not by any measure be the work of informed people. Any ploy would not close down other options with such certitude. To now follow another path would result in a significant loss of face. That is not how politicians tend to operate. They really don't know what they are doing.