But you lose the pro-gnome crowd when you take them out (as you point to in this post). Putting them back in, I don't think that loses the anti-gnome crowd. Like I didn't like Dragonborn or Tieflings, but it wasn't the thing that made me walk away from 4E (I could easily not include those in my setting). And now, as much as I don't like them, I think taking them out would probably be a mistake for the reason you site here (you lose the people who want those in there games). Gnomes might not be as popular but they still have their defenders.
Every option has its defenders. Kobolds, for example, are quite popular and fill a similar role, due to their reputation as sort of Starscream-adjacent dragon servitors/slaves. That it has defenders is not really a reason to include or exclude something from the initial books.
(As an aside, in 4e, ALL books were core, there was no distinction between books in that sense, so 4e gnomes WERE core. They just didn't come out until the second book. Notably, neither did Druid, Barbarian, Bard, Sorcerer, and a handful of other classic options. So it's not like they were relegated to the book equivalent of Siberia. They just didn't get included in the
very first book. People took that as a rallying cry against it, not because they actually cared, but because it was a convenient canard...as was pretty typical of the responses to 4e at launch.)
Dragonborn, by comparison, are by far one of the most popular non-human options in 5e, and their popularity has done nothing but grow since 5e launched. Excluding humans (which are always the most popular race by a wide margin), dragonborn are either 3rd or 4th, depending on whether tieflings have overtaken them again (lumping together all subraces): the nonhuman popularity list is always elf, half-elf, dragonborn, tiefling. Sometimes elf and half-elf swap, and earlier in 5e's life dragonborn were less popular than tieflings. But in general those four have always been very popular and dragonborn specifically have repeatedly grown in popularity.
Popularity is one factor in deciding how important it is to include an option in a publication. They cannot include everything. Would kobolds or goblins be more effective than gnomes as a PHB option? Pathfinder has made their goblins an iconic thing (though I have my issues with that portrayal as well.) It certainly isn't the only factor (otherwise, based on one of the polls they conducted during the Next playtest, we would have gotten a Warlord in 5e and Druid would have been folded into Cleric!) But it is a major factor.
Would the loss of a comparatively small number of gnome fans be mitigated by the gain of, say, a contingent of kobold fans? That's exactly what "man on the street" answers help designers to determine.
1 Eccentric absentminded elder,
2 quick, stealthy watcher
3. Boisterous, obsessive tinkerer
4 licentious lurking caliban
As others have said, 1 and 3 are the same character, just one is old and the other is not; I don't consider age alone to be a personality difference. (Plus, "tinker" is not a personality, it is a hobby or a profession, depending on how strict you define it.) 2 is not even a personality in the first place. "Caliban" is a proper noun, being a specific character, so I'm not sure what you mean by it here, nor do I see anything that really relates to being a gnome in this, as opposed to 1-and-3 which is at least drawing on both classic and recent gnome tropes and ideas (Tolkien's Noldor were similarly "tinker"-y/interested in how things are made, and as others have said we're straight up called "gnomes" at one point.)
So...one and a half personalities is not really doing you much here.
1 is an alchemist possibly an illusionist
2 yes quick and stealthy is a personality, watchful maybe a better term than watcher
3 sure, its typical young gear head, for good measure
4 goblins may be lurking but mine are more malicious than licentious
1. How was that communicated from what was written? I didn't get even a hint of alchemist, let alone illusionist. Someone is an elder by being old. That's the only qualification you need to be an elder.
2. I'm sorry, but no, none of those are personalities, nor do they collectively constitute one. Quickness is a physical trait, as is stealth. Watcher is a profession (e.g. guard) or a hobby (e.g. birdwatching), and "watchful" is a single character trait. You might as well call "angry" a personality, it would be just as descriptive, aka not really descriptive at all. You have described a character in terms of two physical attributes and one personality trait. That's...just not enough to be a personality, nowhere near something like "eccentric absentminded elder," where all three (even the third, despite not being a personality trait) tell us a great deal about the values and mannerisms of the character.
3. Then it's...not a distinct option. At best you've given three, not four. (And, as stated, I think you don't even have that many.)
4. That's not a rebuttal. If anything, it's an admission that the archetypes of gnome and goblin are so similar, it's difficult to distinguish them, which is a strange thing for you to admit, given you seem to be arguing against that point elsewhere.