Nothing is stopping you from making a suggestion telepathically

While suggestion has a verbal component to it, nothing in the description of verbal components nor the spell itself says the verbal component is the suggestion.

Therefore, as long as you are able to chant the mystic words, you can telepathically make the suggestion.

Verbal components (SRD pg. 101):

Most spells require the chanting of mystic words. The words themselves aren’t the source of the spell’s power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion. Thus, a character who is gagged or in an area of silence, such as one created by the silence spell, can’t cast a spell with a verbal component.

Suggestion spell (SRD pg. 181):

You suggest a course of activity (limited to a sentence or two) and magically influence a creature you can see within range that can hear and understand you.

Monsters only

You are quoting a passage from the Monster Manual that specifically talks about Monsters within it. There is no reason to apply those telepathic abilities to a PC without an equivalent from the DMG or PHB or another character based reference.

In addition, the Sage Advice clarification on Awakened Mind states:

In contrast, the telepathy ability that some monsters have (MM, 9) does make two-way communication possible.

This clearly differentiates the Monster telepathic ability from others.

Responding to Monster telepathy

Yes, a PC can respond to a Monster using telepathy as stated in the quote provided in the question (MM, 9)

A creature without telepathy can receive and respond to telepathic messages but can't initiate or terminate a telepathic conversation.