It was compared to everything from a bike helmet to a potty, and allowed to spiral to almost double its budget, before being unceremoniously scrapped and declared a national embarrassment. Zaha Hadid’s futuristic design for Tokyo’s 2020 Olympic stadium would have swaggered through the historic Meiji Park like a glossy white stormtrooper, breaching local building codes and leaving the city with a costly white elephant. But are the alternative proposals, finally unveiled this week, any better?

After Hadid’s heady vision, the two new designs might seem a bit bargain-basement. One looks like an undercooked fried egg – a wobbly white roof with a gelatinous, albumeny middle. The other looks like a pile of salad plates cleared away before anyone had finished, with bits of lettuce poking out from between the stack of saucers.

But both are significantly slimmer and cheaper than Hadid’s design – around 153bn yen (£835m), as opposed to 252bn yen (£1.3bn) – and they appear to be models of a lighter-weight, low-key approach that would fit better with the parkland setting. Anonymously released as Design A and Design B, but believed to be the work of celebrated Japanese architects Kengo Kuma and Toyo Ito, both are titled Stadium in a Forest and make prominent use of wooden construction, in contrast to Hadid’s hefty steel arches.

Scheme A (which looks like the work of Kuma to me) has a roof supported by a dense lattice of exposed timber trusses. If well-detailed it could be a magnificent thing, forming a cat’s cradle of interlocking beams above spectators’ heads and recalling the intricate complexity of Japanese joinery. It would be a first for any contemporary stadium, which usually rely on steel and stretchy skins of PVC and ETFE, and a welcome nod to the country’s building traditions.

From outside, its seating terraces look like slender piles of plates, supported on rings of raked columns with the air of a traditional Japanese temple, while the whole thing is garnished with greenery. It’s too early to tell whether it will be token sprigs or full-sized trees (as some views suggest), but it could enable the park to reclaim the structure after the Olympics, engulfing it with greenery.

Scheme B looks more like it was born on Planet Ito. It has the Pritzker prize-winning architect’s trademark amoebic geometries, with an undulating white steel roof that ripples over the seating bowl, like a toned-down version of SANAA’s entry to the original stadium competition. A colonnade of majestic timber columns march around the perimeter, supporting the roof that flares out like a peaked sailor’s cap. The whole thing is proposed to be sheathed in a full-height skin of glass, bringing all the usual metaphors of transparency and reflection, and allowing the structure to “dissolve into the park”.

Whether covering a stadium in plants or glass offers more potential for it to disappear is debatable – and perhaps irrelevant, given that any Olympic stadium will be very visible indeed, no matter what rhetorical flourishes the architects try and achieve. Design B could be ethereal, or it could look like a lumpen IMAX cinema. Design A could be a wondrous wooden tree-house, or a half-baked idea disguised with garden trellis.

Local reaction has been lukewarm so far. “Both designs have a mundane appearance,” Yasuhiro Kimura, a 34-year-old out jogging near the construction site on Monday, told the Japan Times. “I can’t tell one from the other.” “They don’t have the characteristics particular to Tokyo,” added 16-year-old Akihiro Mori, a high school baseball player, in a disapproving tone.

Such reactions hit on the eternal existential nub of any Olympic stadium design: people expect them to be dazzlingly iconic, the pride of the nation capable of seducing the global TV audience at the opening – yet they also insist they must be built at a cut-price cost with as little impact on the context as possible. The two rarely go hand in hand.

Both Kuma and Ito are eminently capable. They understand the Tokyo context – and they both signed the petition to halt Hadid’s plan, lending the latest competition an unfortunate backstabbing air. When Hadid’s scheme was first unveiled, it was met with a blistering assault from the Japanese architectural community. Arata Isozaki, architect of Barcelona’s Olympic stadium, described Hadid’s project as a “monumental mistake” and warned it would be a “disgrace to future generations”. In a lengthy open letter to the Japan Sports Council, the government body in charge of plans for the 2020 games, he railed against the “distorted” process that had led to “a dull, slow form, like a turtle waiting for Japan to sink so that it can swim away”.

Ito was more measured in his criticism, choosing to quietly work up an alternative proposal that would have incorporated the original 1964 Olympic stadium, a 54,000-capacity venue already on the site. He argued it could be upgraded and reused, just as successful refurbishments had been made to the Olympiastadion in Berlin, host of the 1936 Games, and to the Memorial Coliseum in Los Angeles, Olympic host in 1932 and 1984. His alternative could have been the best option of all – if the structure hadn’t already been reduced to a pile of rubble.

As for Hadid, her practice is bitterly disappointed at the way they have been treated. They had always maintained that the spiralling costs were a result of rising construction costs in Tokyo, not their design.

“It is disappointing that the government did not even consider working with the existing design team to build on the two years of design work they and the Japanese people had invested,” said a spokesman for the practice. “The rules of entry to the new competition restricted the existing design team, as well as many other Japanese and international architects and contractors that wished to take part, from entering. There are now serious risks of a rushed process, with no certainty on the likely construction costs of the stadium, and that it may not be ready in time or deliver a significant sporting legacy without expensive conversion after the 2020 Games.”

The risks are real, and the whole process has been a sorry but familiar saga. Originally intended to host the 2019 Rugby World Cup, the new stadium now won’t be ready in time. But, whichever option they choose, I’m looking forward to seeing what could be the first wooden Olympic stadium in history. As long as they go easy on the fireworks.