Theresa May’s minority government has dropped key Tory manifesto pledges, including expanding grammar schools and a free vote on restoring foxhunting, as the Queen announced a pared-down legislative programme focused on delivering Brexit.

As she formally opened what the embattled prime minister hopes will be a two-year parliament, the Queen set out the government’s intention to deliver the eight bills necessary for Brexit – including legislation allowing Britain to determine its own immigration, customs and trade arrangements.

The Queen was accompanied by the Prince of Wales in the traditional ceremony in the House of Lords, after Buckingham Palace announced that the Duke of Edinburgh had been admitted to hospital as a precaution on Tuesday night.

The speech included a series of domestic policies, including sweeping changes to technical education, establishing a new statutory anti-extremism commission.

But grammar schools, one of May’s flagship proposals in the early weeks of her premiership, were not mentioned, despite the Conservative manifesto pledge to consult on opening new selective schools.

Instead, the Queen said: “My government will continue to work to ensure that every child has the opportunity to attend a good school and that all schools are fairly funded. My ministers will work to ensure people have the skills they need for the high-skilled, high-wage jobs of the future, including through a major reform of technical education.”

With different cabinet ministers striking markedly different tones since the election result about what sort of Brexit deal they expect Britain to strike, and May hemmed in on both sides by MPs in her own party, there was little detail about the nature of the immigration, trade or customs policies the government will seek to implement.

The government promises to “work to improve social care” and “bring forward proposals for consultation” rather than implementing the highly controversial policy set out in the manifesto. And the promise of a free vote on restoring hunting with hounds, which enraged some young voters, has also gone.

The speech also appeared to confirm that Donald Trump’s planned state visit to the UK has been postponed for the foreseeable future.

The Queen’s speech traditionally lists any upcoming state visits for the parliament, and she mentioned the planned visit in July by the king and queen of Spain. However, there was no mention of Trump coming, following reports he might not want to come if this prompted protests.

The prime minister, who is yet to conclude a deal that would see the Democratic Unionist party’s 10 MPs lend ongoing support to her agenda, is keen to show that she can continue to govern, despite seeing her majority wiped out at the 8 June general election. But the broad outlines of the speech – originally scheduled for Monday – were agreed with the DUP last week.

MPs will be asked to support the speech in a vote next week – and opposition parties are likely to table a series of amendments.

The government plans to press ahead with changing the funding formula for schools, a measure which has been criticised by a number of Conservative MPs.

More than 9,000 schools in England were expected to lose funding and those awarded extra funding have warned the gains are likely to be outweighed by real-terms funding cuts. However, after the criticism the Conservative manifesto promised the government would “make sure no school has its budget cut as a result of the new formula”.

The detail in the Queen’s speech commits the government to the changes, saying the data used to allocate funding to local authorities is “over a decade out of date” and changes had been “widely welcomed across the sector” despite widespread criticism from MPs and teaching unions.

“That is why we recently consulted on a national funding formula for schools and why we will deliver on our manifesto commitment to make funding fairer,” the government’s briefing says.

Instead of grammar schools, improving technical education was front and centre of the commitments in the speech, promising an investment of an extra half a billion in England’s technical education system and plans for new Institutes of Technology.

The Queen said the government would also seek to introduce a series of pro-consumer measures, seen by ministers as both popular with voters and likely to pass the Commons.

This includes a planned travel protection bill, intended to revamp consumer regulations in the era of internet-booked holidays, and a long-promised law to crack down on compensation claims for people claiming whiplash injuries following car crashes.

Another law will ban landlords or agents from charging fees to tenants seeking to rent properties in England – a pledge first made last autumn.

Among planned changes to the legal system is a proposal to allow people charges with some low-level offences to plead guilty and pay a fine online.

There will also be a new data protection bill, as outlined in the Conservative manifesto, giving people the right to seek social media companies to delete certain information about them, as part of wider measures to guard people’s data online.