1. The Issue

The telepathic feat doesn't seem to be teaching the character the detect thoughts spell. It does allow the character to use the spell once per long rest, but doesn't allow the character to cast the spell using spell slots. And the once per long rest use still states it has verbal, somatic and material components.

The third bullet point of the telepathic feat states:

• You can cast the detect thoughts spell, requiring no spell slot or components, and you must finish a long rest before you can cast it this way again. Your spellcasting ability for the spell is the ability increased by this feat. If you have spell slots of 2nd level or higher, you can cast this spell with them.

1.1. Spell Components

Telepathic feat, bullet 3, part 1:

You can cast the detect thoughts spell, requiring no spell slot or components, and you must finish a long rest before you can cast it this way again.

When a character gets the telepathic feat they gets this special use of detect thoughts in their spell list

Text Version in case image link goes down:
USE Detect Thoughts <c> Telepathic (Ability)   1A    Self    WIS DC    Social*        1/LR, D: 1m, V/S/M

Despite the feat text stating that the special cast that is available once per long rest does not require components, the spell added to the character's spell list still denotes the verbal, somatic and material component requirements (V/S/M at the far right).

1.2. Cast with Spell Slots

Telepathic feat, bullet 3, part 3:

If you have spell slots of 2nd level or higher, you can cast this spell with them.

When a character gets the telepathic feat they do not learn the detect thoughts spell, so just with the feat (without learning the spell another way) the character cannot cast detect thoughts using their spell slots.

2. RAW vs RAI

I understand that RAW the telepathic feat does not state that you learn the detect thoughts spell, so one could argue that the third clause of the third bullet point of the telepathic feat that says that you can use spells slots to cast detect thoughts is only valid if you learn detect thoughts from another source and is used only to counter the first clause saying you must finish a long rest before you can cast it [...] again.

But my understanding was, prior to using it in the D&D Beyond Character Builder, that the character did indeed learn the detect thoughts spell upon acquiring the telepathic feat, and that it gave the character a special use of the spell once per long rest, and also allowed the character to cast it normally using spell slots.

3. The Question

Was the telepathic feat intended to give you the detect thoughts spell for free and there is a problem with the feat code in the D&D Beyond Character Builder or was it only intended to give you a specific use of the detect thoughts spell without giving the spell for free?

1. Detect thoughts can totally be cast with spell slots, this is RAW. However, D&D Beyond doesn't add it to your list of spells that can be cast with slots based on feedback on such things cluttering up the UI/UX. As such, if you want to cast it using a spell slot, you just mark off the spell slot as used directly
2. The system currently doesn't have a way of modifying spell listing information on the character sheet based on additional modifiers, such as not using components. That is why the spell you see still lists the components; it's providing the full spell information, unmodified. You have to remember, from reading the description of the feat, that you can ignore that information.