It is two years since Brendon McCullum, freshly appointed as coach of England’s Test team, said he had pursued that job over a similar one with the white-ball side because in limited-overs cricket “the team is flying” and “I wasn’t interested in a cushy kind of gig”. With that position vacant once more after the sacking of Matthew Mott, it is safe to say that nobody will be turning it down for similar reasons this time.

The team are underperforming, ageing and overdue a comprehensive overhaul. The good news is that this is as close as you can get to the ideal time for one: after three World Cups in less than two years there is now a gap when the only international tournament is the Champions Trophy, a second‑tier event scheduled to be played in Pakistan in early 2025, before three World Cups and the sport’s Olympic return are crowbarred into two and a half years between February 2026 and October 2028.

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The good news for the new coach is that England’s strongest team should be a match for any in the world; the bad news is that he may never be able to pick it.

In recent years, England’s schedule has tipped from stuffed to bizarre: factor in two days’ training before each Test, one for each white-ball game, a day’s recovery after each, plus a day for each international flight or domestic venue change and of the 123 days between the start of their next game and the end of their tour of New Zealand in December they will have teams on duty on 142.

The idea of McCullum taking over the white-ball side without abandoning the Test team, as proposed in the past week by the former captain Eoin Morgan, dies on first contact with the schedule. “We can’t put our best team on the park,” Mott said last year. “We’ve just got to find a way to compete.”

The role the England and Wales Cricket Board’s managing director of men’s cricket, Rob Key, is now attempting to fill involves lots of quandaries and compromises. It also pays less than a head coach gig at the IPL, which involves around two months’ work a year.

So he may struggle to tempt Kumar Sangakkara, the genial Sri Lankan who coaches Jos Buttler’s IPL franchise, Rajasthan Royals, but he is now based in Dorset, has an excellent relationship with the captain and will surely be sounded out.

Other outstanding and handily located foreign candidates include Stephen Fleming, like Sangakkara an experienced and widely admired IPL head coach, who is at Southern Brave in the Hundred, and the New Zealander’s batting coach at Chennai Super Kings, Mike Hussey, who has worked with Buttler a couple of times, most recently at the 2022 T20 World Cup, and is at Welsh Fire.

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Meanwhile, at Northern Superchargers, in his first senior coaching role, is Andrew Flintoff, the bookmakers’ favourite, who in March was described by Key as “a worthy candidate” who anyone in his position “would be stupid not to look at”. But the 46-year-old would be a ludicrous appointment in many ways: he is charismatic and the players clearly enjoy having him around, but his lack of coaching experience, his close friendship with Key and his association with the ECB chairman, Richard Thompson, who also chairs the agency that represents him, should put an end to his prospects.

If an English coach is sought, James Foster has better credentials: he is two years younger and vastly more experienced, though Flintoff got his Hundred gig only because the team wanted a new face after finishing last under Foster previously.

Morgan is a natural leader who has guided England through a successful process of renewal before, but has said that “the timing is not right” for his return and is surely correct – he also lacks coaching experience and, having led most of them to victory in the 2019 World Cup, is too close to the players who must now be left behind. Bad timing might also eliminate Rahul Dravid, given it is less than a month after he stepped down as India’s head coach. On the other hand, perhaps this is the ideal moment for Jonathan Trott, Afghanistan’s head coach, who is known to covet the role and whose star is in the ascendant.

Last year, he described his career plan as “hopefully having a very, very good World Cup and then seeing what the options are”. Since then he and Afghanistan have had two superb World Cups, beating England in one of them. Trott’s contract ends in December, though he has just agreed to lead Pretoria Capitals in January’s SA20 and is considered an outsider in this race.

Then there is Marcus Trescothick, the caretaker, whose singular virtue is incumbency. Should Key not be able to attract the first couple of names on his list, and England get a few good results when they face Australia next month, there will surely be a temptation to just let it flow.

Key’s job, though, is to look beyond easy solutions: if England struggle to put their best team on the park, they really need the best team off it.