Mc Allen, a stay-at-home dad who lives in Dogpatch, took his 4-year-old son on an unusual field trip the other day: to a soon-to-open homeless shelter.

Allen, a board member of the Dogpatch Neighborhood Association, persuaded reluctant neighbors not only to accept the city’s newest Navigation Center in their own backyard, but to embrace it.

“When I look out at the weather right now, the urgency couldn’t be more real,” Allen told Mayor Ed Lee and other city officials who toured the center that’s set to open in March and house 70 people. Rain pounded the trailers, and wind whipped the group’s umbrellas. “I wish this place was already open.”

That welcoming attitude is rare, even in a liberal, compassionate city that professes to want to help its huge homeless population. Hand out change, give money to nonprofits, volunteer in a soup kitchen? You betcha. But accept a shelter or service provider in your neighborhood? That’s another story, even among supervisors tasked with solving the city’s most frustrating problem.

Lee is adamant that Navigation Centers and homeless services spread beyond the Tenderloin and into less likely parts of San Francisco, such as the site at 25th and Michigan streets in Dogpatch. But the mayor has already tripped up in finding supervisors willing to go along with the plan.

He said in two separate interviews that he asked candidates for the District Eight seat, vacated when Scott Wiener won a seat in the state Senate, if they would accept a Navigation Center in the district and that they had to say yes to get the job. But his hand-picked choice, Jeff Sheehy, said that while they discussed Navigation Centers, Lee never asked him whether he’d back one — and he won’t.

“I don’t think it would be appropriate in District Eight,” Sheehy said. “If you want to have a Navigation Center, there should be services nearby, and you should have neighborhood support. I don’t see a site in District Eight where that would be possible.” His district includes the Castro, Noe Valley and Glen Park.

Sheehy is in his first week on the job — distancing himself from the mayor sure didn’t take long.

But Sheehy’s attitude is common — among supervisors and San Franciscans at large. While the new Dogpatch center breaks the mold, the last Navigation Center at the Civic Center Hotel and the next one in the Salvation Army building at Sixth and Jessie streets, set to open by the end of the year, are in more predictable locations.

Statistics from an internal document prepared by the Department of Homelessness and Supportive Housing and obtained by The Chronicle show just how concentrated services are in District Six, which includes the Tenderloin and much of SoMa. It also shows that other districts are not shouldering their weight.

While the numbers are a little dated — they’re from 2015, when the last citywide homeless count was conducted — the overall picture hasn’t changed much. District Six has 56 percent of the homeless population, but 65 percent of its shelter beds and 73 percent of its permanent supportive housing.

District 10, which includes Bayview-Hunters Point and Potrero Hill, appears to be have the biggest shortfall. According to the city, it has 17 percent of San Francisco’s homeless population but just 8 percent of its shelter beds and 4 percent of its permanent supportive housing units.

The numbers don’t count the coming Dogpatch shelter beds, but the Bayview neighborhood did infamously battle the city over a proposed shelter on Jennings Street, and the city scrapped its plans in 2015.

The Bayview neighbors aren’t alone. Supervisor Mark Farrell has had a laser-like focus on homelessness, but there’s still almost nothing for homeless people in District Two, which includes the Marina and Pacific Heights. (And yes, there are homeless people there: 60 by the last count.)

A recent proposal by Community Housing Partnership to demolish the Bridge Hotel on Lombard Street and build 100 units of supportive housing met with major neighborhood resistance — in part because it would have required breaking the zoned height limitations — and didn’t move forward. Farrell didn’t go to bat for the project.

When former Supervisor David Campos circulated a list of potential sites for more Navigation Centers last year, he left off any sites in Bernal Heights, where he lives.

West Portal neighbors were irate, as Campos’ list included two parking lots there. Supervisor Norman Yee said the sites weren’t appropriate because they were near family-oriented businesses. Hmmm, are there no families or kids in the Tenderloin? Asked which sites in District Seven would be appropriate for Navigation Centers, Yee said, “I can’t think of any.”

Yee also doesn’t support a proposed 150-unit apartment complex for low-income seniors, some of whom would be formerly homeless, in the Forest Hill neighborhood. Neighbors have fought the plan, and a lawyer representing them said it should be relocated “somewhere along the Van Ness corridor next to a Holiday Inn.”

These attitudes need to change, Lee said. For the record, he lives in District Eight in Glen Park and said he’d be “very willing” to establish a Navigation Center in the district. (District Eight has 5 percent of the city’s homeless population, but 3 percent of its shelter beds and 1.5 percent of its supportive housing.)

“Everybody in every district and every supervisor needs to contribute to this idea,” Lee said. “When people get used to helping transform people’s lives — and not just watching it happen in a negative way on the streets — they’re going to be a lot more enthusiastic.”

It may help that Navigation Centers aren’t your standard scary-looking homeless shelters. They’re intended to be more welcoming than traditional shelters and offer intensive case management, storage, come-and-go privileges and the ability to sleep next to partners and pets.

The Dogpatch center has the look of a cheery elementary school: bright green and blue trailers that are raised off the ground and will be connected by elevated walkways.

City officials who discussed the plans, which include trailers situated around community courtyards with picnic tables and greenery, used oh-so-San Francisco phrases like “outdoor dining options” and “a restorative environment” in a “village concept.”

“This is the classiest Navigation Center we have thus far,” Lee said.

Whether he can convince other neighborhoods that the centers can class up their areas rather than destroy them remains a big question.

San Francisco Chronicle columnist Heather Knight appears Tuesday and Friday. Email: hknight@sfchronicle.com Twitter: @hknightsf