I don't think we should underestimate the desire for balance here. Even if the starter Pokémon may not directly face each other so often in-game that balance becomes a gameplay issue, people will still compare them to each other, and it's bad (for merch sales if anything) if one of them stands out as clearly better or worse than the other two.
Take the triangle Grass/Ground/Poison, for instance. The Grass starter needs to be the Grass type, obviously. If the Water starter gets Poison as a secondary type, the Grass starter no longer has any advantage over it. Grass has no supereffective STAB vs. the Water starter, while both the Fire and Water starters now have supereffective STAB against the Grass starter --> the Grass starter becomes the objectively worst of them. So the only choice is Water/Ground, which means a double weakness to Grass, while the Fire starter has a double weakness to Ground --> the Grass starter is the only one left without a double weakness to the STAB of the others. Come to think of it, though, some balance would be restored if one starter was Grass/Fairy, creating an interesting triangle of double weaknesses.
Ice/Ground/Rock also works in one configuration. All of the three types have weak match-ups against two of the starter types, but are strong against the last. Grass/Ice, Water/Ground, and Fire/Rock would all hit each other double-effectively. The Grass starter would be the only one to have neutral STAB against the starter it's weak to, however, making the Fire starter a weaker option within the triangle. However, it'd be crazy-good against pretty much everything else the game throws against it, resisting seven types and hitting six super-effectively - conversely, the Grass starter would be weak against everything. The Water starter would have only one weakness.
Electric/Water/Ground is an interesting one. Obviously, Water is fixed here. If the secondary Ground type went to the Fire starter, it would have bad matchups against both the others, while neither of the others would have a disadvantage to compensate. But if we flip it around to Fire/Electric and Grass/Ground, Water gets bad matchups against both the others, while Grass gets good matchups both ways. Water/Flying, Fire/Ground, and Grass/Electric would be an interesting triangle, however - but probably shafting the Fire starter too much, while Grass could duke it out both ways.
I don't see Bug/Grass/Rock working very well. Grass is fixed, so the others would be Fire/Bug and Water/Rock, or Fire/Rock and Water/Bug. Either way, the Water starter gets a 4x advantage over the Fire starter, and Grass either hits both the others neutrally or trounces Water completely.
Rock generally doesn't work well with the starters, because it always creates a 4x weakness or messes with the relative weaknesses and resistances in the Grass-Fire-Water triangle. Fire/Rock is doubly weak to Water (and Ground), Water/Rock is doubly weak to Grass, and Grass/Rock is hit neutrally by the other two types (while hitting both for super-effective damage in return). Something similar goes for Ground, and the opposite for Bug and Normal (they really are the short straw when it comes to types, mostly a disadvantage to whatever is holding that type).
That being said, there seems to be a surprisingly large number of combinations that are "almost" perfect enough that they should be considered. A little bit of going off-balance worked well in generations I, III, and IV.
(except Legends Arceus ofc)
