Though at first glance it might appear, that if the DM does not give out magical weapons, then many monsters in the DM's Basic Rules will make half of the party irrelevant unless the DM gives out magical items and thus the game is dependent on magical items, unlike what the article states. However, a close look at the classes and character abilities reveals that this is not the case.
Firstly, there are two spells, magic weapon and elemental weapon, which will make any nonmagical weapon into a magical weapon. Magic weapon, however, only works on one weapon as it is a concentration spell, and is not available until 3rd level as it's a level 2 spell available to Wizards, Paladins and War Domain Clerics.
At level 6, Monks and Circle of the Moon Druids both get abilities which make their unarmed or natural attacks count as magical weapon attacks.
The Paladin and Warlock at third level gain an ability to have their weapon count as magical - the warlock through the Blade Pact boon, and the Paladin through the Sacred Weapon channel divinity.
Looking over the monsters in the DM Basic Rules, no creature below CR 5 has immunities to nonmagical weapons (meaning these immunities should not be encountered until the party is of a high enough level to deal with them) save the Werewolf; however, the werewolf's immunities can be bypassed by magical weapons or silvered weapons. Silvered weapons can be obtained by spending 100 gold per weapon in just about any location that sells weapons.
So while the Player's Handbook makes it very clear that magic is still a big part of D&D, and magic in general is assumed in the game, the delivery of special magic items from the DM to the players is not. Teamwork, resourcefulness, and character abilities will allow you to overcome any challenge without requiring magic items or making magic items part of the assumed progression of character development.
Of course, as a DM, one should make sure that the party has the necessary information to prepare for battles where immunities of any kind might completely shut down a player.
It is worth noting, that perhaps unlike in other versions of D&D, in 5e the math is used partially to enhance the story, and partially to "fade to the background" while playing the game.