Union leaders have said there is no prospect of industrial action on Southern ending soon, as passengers on Britain’s worst-performing rail network suffered fresh disruption on the first day of a series of three-day strikes.

Overrunning engineering works on the Brighton mainline compounded problems for commuters as rush-hour trains from the south coast were cancelled or delayed by up to an hour.

The owners of the troubled franchise, Govia Thameslink Railway, made a fresh appeal for unions to end their dispute after a last-minute legal challenge failed, but leaders of the Rail, Maritime and Transport workers’ union (RMT) said the company was “peddling lies” and vowed to continue their action. Its members went on strike at midnight in their long-running dispute over proposed changes to the role of conductors.

Southern had been intending to run about 61% of its normal timetable, before the additional problems caused by engineering work. Network Rail apologised for a breakdown that meant the railway between Gatwick and Brighton only reopened to trains at 7.20am. Rush hour at Victoria, Southern’s biggest hub, was quieter than usual as trains failed to arrive, station staff said.

Speaking on a picket line outside the station, Paul Cox, the RMT’s regional organiser, claimed commuters remained sympathetic, despite the disruption. He said: “Based on what I’ve seen today, people taking our leaflets, they don’t blame the union. This is only our 10th day of strike action and they know the reality on Southern – they see through the myth of unofficial industrial action.”

Earlier, Charles Horton, the chief executive of Govia Thameslink Railway, told BBC Radio 4’s Today programme: “I can’t tell you when the RMT are going to stop striking. They should stop striking now. The reasons for the dispute have ended.”

The company claims the union told its conductor members to accept a deal last week. The RMT has advised individual members, who are otherwise threatened with dismissal, to accept new contracts that GTR is sending out to rebrand conductors as onboard supervisors.

Horton apologised to passengers for what he accepted had been an “extremely challenging period” for Southern.

He said: “I’m incredibly sorry for passengers who have suffered months of misery as a result of the ongoing industrial action. The railway on this part of the network is going through a huge upgrade programme to tackle some of the longer-term issues which need to be resolved here in terms of providing more capacity, introducing new trains and improving the punctuality and reliability of services.”

He added: “We also need to modernise the way in which our staff deliver service for customers and that is what is at the heart of this dispute here.”

The RMT’s general secretary, Mick Cash, said: “Once again this morning Southern have been peddling the lie that this dispute is about who opens and closes the doors when, in fact, it’s about the retention of a guaranteed second safety-critical member of staff, with the full range of safety competencies, on all current services. Despite all the spin and bluster, the company refuse to give that guarantee.”

Paul Maynard, the rail minister, said: “It is disappointing that passengers once again face needless and unjustified strike action by the RMT, after the union advised its members to accept the new roles being offered by the operator. The union leaders have continually rejected a deal that protects jobs and ensures that conductors will carry on delivering safe, accessible and more reliable rail services. I call on RMT to put passengers first and stop this damaging action.”

But at Victoria, Cox warned: “The strikes are just going to continue until they come to their senses. They’ve put a hardline management in place and we’ve just got to break them down.”

Some Conservative MPs have called on the government to take action, but Chris Grayling, the transport secretary, is believed to be reluctant to get directly involved, believing a direct public confrontation would only benefit the RMT. But more rail franchises are expected to mandate fewer staff on trains to reduce costs.

On Monday the Rail Safety and Standards Board and Southern were forced to reiterate the safety organisation’s independence after the RMT circulated a briefing memo showing Horton’s links to the RSSB and a report that discussed the economic benefits of making guards redundant.

The row over the safety role of guards looks set to continue throughout the railway, underlined on Tuesday as the RMT made a fresh demand that the Merseytravel authority abandon “reckless” plans to introduce driver-only trains on Merseyrail.