Drupal - How do I programmatically remove one breadcrumb link?

Drupal 7 and Drupal 8 are more or less the same in some aspects.

If you use preprocess_breadcrumb in your theme, you would do

function my_theme_preprocess_breadcrumb(&$variables) {
  array_pop($variables['breadcrumb']);
  ...
}

You could also solve this in the template file, by skipping the first item in {{ breadcrumb }}

If you want module only solution, you could swap the breadcrumb service with your own thing.

If you are using a BreadcrumbBuilder, and try to remove a Link already added, you can't. Unless you create a custom Breadcrumb class, to add a new method or override an existing one.

My solution was this :

<?php

namespace Drupal\my_module;

use Drupal\Core\Breadcrumb\Breadcrumb;

/**
 * Blablabla.
 */
class BreadcrumbCustom extends Breadcrumb {

  /**
   * Sets the breadcrumb links.
   *
   * @param \Drupal\Core\Link[] $links
   *   The breadcrumb links.
   * @param bool $force
   *   Boolean to indicate if you should force to override the existing links.
   *   Use with caution.
   *
   * @return $this
   *
   * @throws \LogicException
   *   Thrown when setting breadcrumb links after they've already been set.
   */
  public function setLinks(array $links, $force = FALSE) {
    if (!empty($this->links) && !$force) {
      throw new \LogicException('Once breadcrumb links are set, only additional breadcrumb links can be added.');
    }

    $this->links = $links;

    return $this;
  }

}

and inside the BreadcrumbBuilder :

$links = $breadcrumb->getLinks();
unset($links[1]);
$breadcrumb->setLinks($links, TRUE);