The Confederation of British Industry has urged Theresa May to drop her “blunt target” on immigration numbers and introduce new freedom of movement rules for EU citizens post-Brexit to ensure firms, large and small, can stay in business when the UK leaves the bloc.

Outlining the results of a major consultation with business leaders, Josh Hardie, the CBI’s deputy director general, said companies believed an injection of honesty was urgently needed in the political debate about migration.

“This is no longer a theoretical debate, it’s about the future of our nation. False choices and sloganeering must be avoided at all costs,” he said.

In a report, published on Friday, the CBI argues that the nation’s needs are more complex than simply ensuring that the UK could attract the “brightest and best”.

It calls for new rules for EU citizens to keep open the pipeline of migrant workers in all sectors including agriculture, hospitality, construction, the NHS and the creative industries; and an easing of the policy for non-EU workers to give small businesses, unable to afford the visas or deal with Home Office red tape, a chance to plug any gaps arising from Brexit.

“Openness and control must not be presented as opposites. Public attitudes towards migration and the impact it has on communities are far more nuanced,” said Hardie.

The CBI’s report on immigration comes weeks after a cabinet split on post-Brexit immigration policy, with the former home secretary Amber Rudd’s plans to give EU citizens preferential treatment reportedly scrapped in June by her successor, Sajid Javid.

The CBI report – Open and Controlled, A New Approach to Migration – concludes that a root and branch change is needed in Britain’s immigration approach following what it says was an “extensive” consultation with employers and trade associations representing 124,000 firms.

It argues that May’s long-held immigration target of 100,000 migrants a year is not viable; that rules on visas for non-EU workers are too expensive and too restrictive; and a new EU citizens-only policy needs to be developed to keep the economy on the road.

“Scrapping blunt targets, ensuring all who come to the UK contribute and using the immigration dividend to support public services will add to public confidence,” said Hardie.

“The building blocks of a successful new migration system for the UK begin with an honest and open debate that has been absent from politics. The stakes couldn’t be higher. Get it wrong, and the UK risks having too few people to run the NHS, pick fruit or deliver products to stores around the country. This would hurt us all – from the money in our pockets to our access to public services.”

While recognising that immigration “has also given rise to legitimate public concern about the pressure it creates on public services and on society”, the report says leaving the EU will mean “momentous change” for business which should be addressed properly by the government.

Shifting the tone of the debate to focus on the positive benefits of migration will help build public trust, it says.

The CBI argues that immigration has “delivered significant economic benefit to the UK” over the past 50 years and maintaining access to people and skills was “a high priority for business in the UK as it prepares to leave the EU”.

The report reflects findings from research and surveys of 18 different sectors, one of the largest consultations of its kind. It says a pragmatic approach to migration will secure Britain’s future as a country open for business and foreign-inward investment.

The CBI argues that simply extending the current approach to non-EU citizens to all migrants will be unworkable for business as the current tier-2 visa system involves a cap on numbers coming to Britain each month and is tilted in favour of higher earners.

In April, Danny Mortimer, the chief executive of NHS Employers, revealed he knew of at least 400 doctors who had been blocked from entering the UK to take up job offers because of the cap.

It recommends registration for all EU citizens migrating to the UK following Brexit and to restrict their stay to three months unless they can prove they are working, studying or self-sufficient.

This mirrors existing freedom of movement rules under EU law but would bring the UK into line with most other nations within the bloc by giving countries the means by which to monitor migration through registration.

It also recommends that the government should reinstate controls on access to in-work benefits but continue to exempt EU workers from a health charge that applies to non-EU workers.

It also wants the government to ringfence the rights of EU citizens to secure their place in the country in the event of no deal, echoing calls from campaigners.