As usual, Ms May is right in there with the misinformation from the start
"Her logic, she said, was that UK had voted overwhelmingly to end open borders"
This is simply not true.
Firstly, the UK voted in a referendum by 51.9% to 48.1% to 'leave the EU'. This is hardly 'overwhelming', especially when you consider the turnout, and those who were ineligible to vote.
Secondly, 'leaving the EU' is not 'ending open borders'. People voted 'out' for a myriad of reasons, only one of which was having open borders with the rest of Europe.
Furthermore, it is worth reiterating that the referendum was non-binding, by design. In the briefing paper that all MPs received before the vote to have the referendum, this was explicitly pointed out, along with a proviso that if any vote were to be binding then it would need a 'supermajority' of 66% of the vote (and a minimum turnout) in order to be binding.
It is in the job description of every MP to read and understand the briefing papers on commons votes before voting on them, so none can claim to not know this, and actually be doing the job they are paid for.
The briefing paper for the referendum bill can be found here.
Section 5 explicitly states:
[the bill] does not contain any requirement for the UK Government to implement the results of the referendum, nor set a time limit by which a vote to leave the EU should be implemented. Instead, this is a type of referendum known as pre-legislative or consultative, which enables the electorate to voice an opinion which then influences the Government in its policy decisions.
Section 6 discusses thresholds in referendums, and what would be expected if the referendum were to be binding (which it isn't).