This post has been revised due to a ruling by Jeremy Crawford. See the revision at the bottom of this answer, which has been upheld by the ruling.
Original answer: using the second point probably shouldn't prevent the first point from being used later.
That the phrase "a creature can't regain hit points from this feat again until it finishes a short or long rest" is in the second bullet point and not affixed as an addendum after both bullet points strongly suggests that it is meant to apply only to the second bullet point's healing functionality, not the first bullet point's stabilizing functionality.
In other words, I believe the phrase is implied to mean "a creature can't regain hit points in this way again until it finishes a short or long rest," although the wording may not have been as accurate as it could be to reflect that.
This interpretation is reasonable in a thematic sense: it makes sense that the kit could still be used to stabilize a creature to 1 hit point yet no longer be able to heal a person beyond what their body's own rest is capable of.
Although I strongly believe that this is a reasonable ruling for a DM to make, if you want to be as literal and pedantic as possible in the interpretation of the rules as written then you must indeed prevent all subsequent healing through the use of the feat for either bullet point once the second bullet point has been used, and therefore the second bullet point would indeed nullify the first.
I really doubt that is the intention, however, given the organization of the phrase and bullet points and the apparent thematic motivation, but I can no find no official word clarifying the intent of the rules for the Healer feat in any official Tweets or Sage Advice (see revision below which nullifies this). At the very least, I think the DM is justified in ruling either way.
Revised answer: using the second point doesn't prevent the first point from being used later.
See the following clarification by Jeremy Crawford (see Twitter), the lead rules designer for 5e.
Question (paraphrased): Is the second feature of the Healer feat intended to work on/revive unconscious characters? If so, what's the benefit of the first feature? Just that it doesn't prevent the regaining of further hit points like the second feature does?
Answer: Yes, and yes.
Regardless of how we read the bullet points, the intention is that the first bullet point can still be used on a creature even after the second bullet point has been used on them. This upholds the original answer in this post.