Yes, attunement is only required for the beard growth
The relevant rules on attunement state that certain magic items can give you some of their magical properties without attuning to the item (DMG, pg. 138)
Without becoming attuned to an item that requires attunement, a creature gains only its nonmagical benefits, unless its description states otherwise.
Since the Belt of Dwarvenkind simply states that certain properties are gained whilst wearing the belt, but other properties are called out as requiring attunement, this falls under the "description stating otherwise", meaning that in this case, specific beats generic.
For comparison, there are certain weapons that allow you to use it as a +X weapon without attuning to it, but has other properties that become apparent once you attune to it. Consider the Hammer of Thunderbolts (DMG, pg. 173):
Hammer of Thunderbolts
Weapon (maul), legendary (requires attunement)
You gain a +1 bonus to attack and damage rolls made with this magic weapon.
Giant's Bane (Requires Attunement). You must be wearing a belt of giant strength (any variety) and gauntlets of ogre power to attune to this weapon. The attunement ends if you take off either of those items. While you are attuned to this weapon and holding it, your Strength score increases by 4 and can exceed 20, but not 30. When you roll a 20 on an attack roll made with this weapon against a giant, the giant must succeed on a DC 17 Constitution saving throw or die.
The hammer also has 5 charges. While attuned to it, you can expend 1 charge and make a ranged weapon attack with the hammer, hurling it as if it had the thrown property with a normal range of 20 feet and a long range of 60 feet. If the attack hits, the hammer unleashes a thunderclap audible out to 300 feet. The target and every creature within 30 feet of it must succeed on a DC 17 Constitution saving throw or be stunned until the end of your next turn. The hammer regains 1d4 + 1 expended charges daily at dawn.
And the Berserker Axe (DMG, pg. 155):
Weapon (any axe), rare (requires attunement)
You gain a +1 bonus to attack and damage rolls made with this magic weapon. In addition, while you are attuned to this weapon, your hit point maximum increases by 1 for each level you have attained.
Both of these mention being a +1 weapon, but then go on to say how it gives you more if you are attuned to it (although for the former, this is all under the subheading "Giant's Bane", which the Belt of Dwarvenkind doesn't have, but it's still evidence of a magic item that has some magical properties that require attunement and some that don't require attunement).
This supports my interpretation that the Belt of Dwarvenkind gives you everything but the beard growth without requiring attunement, since it at least calls out that which does require attunement, allowing us to infer that the rest must not require attunement.
Note that my answer does not take into account Jeremy Crawford's tweet (because I didn't know about it when I wrote this), and I am therefore simply going to leave this answer as-is to show my reasoning in absence of this JC quote, based on how the Belt of Dwarvenkind's description is written. (I could do a U-turn, but then I'm just duplicating Sdjz's answer, so this is probably more useful left as an alternative set of reasoning).