One respectful follow-up is completely fine. Busy consultants miss messages all the time, and a short nudge after several days is normal professional behavior, not desperation.
The tone matters, though. Keep it light, acknowledge that they may be busy, and make it easy for them to decline or ignore without creating awkwardness. If there is still no reply after that, move on.
It also helps to widen your funnel. Networking goes much more smoothly when you are not emotionally tied to one specific person responding.
Being left on read does not necessarily mean you did anything wrong. It usually just means the other person had different priorities that week. You may also find it helpful to read networking follow-up strategies and examples if you want a more structured way to practice this.