The d3 has been used for many different mechanics.
This search on DnDBeyond gives all the instances of "d3" appearing in the DDB compendium material, which covers the first three sections here. Note, this one has several false positives, as "D3" is used to denote locations in many adventures. All the other searches I link here use "1d3" to get hits.
The d3 first shows up in the introduction to the Player's Handbook, where it is explained how you can simulate one with a d6:
The same d notation appears in the expressions “1d3” and “1d2.” To simulate the roll of 1d3, roll a d6 and divide the number rolled by 2 (round up).
We now examine many of the ways the d3 has been used.
Falling down a chimney.
In one of the adventures published in Explorer's Guide to Wildemount, you can fall down a chimney:
Any character who tries to descend the chimney without a rope or a climbing speed inevitably falls. Roll a d3 to determine which lower chimney the character tumbles down.
Betting on dinosaur racing and meeting with princes.
In Tomb of Annihilation, you can bet on dinosaur races at one of the major towns. Two of the possible results depend on the outcome of a d3 roll.
In this same town, the party can seek an audience with a merchant prince. The module states:
Characters must wait 1d3 hours before being given an audience.
Having close family relationships.
In the character building options in Explorer's Guide to Wildemount, you roll a d3 to determine how many of your family members you are close to:
Roll a d3. This is the number of powerful relationships you have within your family.
And a plethora of magic items.
This search shows 33 magic items that use a d3 in some manner, most of them are for recharging at dawn. I will not list them all here.
The Ioun Stones use a d3 to determine how far from your head the stone orbits.
The decks of Many Things and Several Things use a d3 to determine the number of times a certain ability can be used.
And a few monsters.
This search gets us the mosnters. Five monsters use a d3 to determine the number of creatures they summon with a particular ability. For example, the Crokek'toeck:
Crokek’toeck opens its mouth and disgorges 1d4 barlguras, 3d6 gnolls led by 1 gnoll fang of Yeenoghu, 6d6 dretches, or 1d3 vrocks.
The Assassin Bug lays eggs inside you:
The target is infested with 1d3 assassin bug eggs, which immediately hatch into assassin bug maggots.
The Olhydra makes it rain:
Violent downpours become frequent within 10 miles of the lair. A downpour occurs once every 2d12 hours, and lasts 1d3 hours.
How many slaves does a Booyahg Goblin have?
This search gets us the goblin playable race. But this is misleading. DDB took the Monster Lore section on Goblins from Volo's Guide to Monsters and put it on the same page with the goblin playable race information. I recommend against giving your players slaves.
Booyahg Whip. Khurgorbaeyag saw fit to gift this goblin with powers that enable it to dominate others. The goblin has 1d3 other goblins that slavishly obey its orders.
And an indeterminate amount of times for random encounters.
This search gives an overflow of results at DnDBeyond. I'm not going to count them. Most are random encounters from Xanathar's Guide to Everything, but there was a hit for Tales from the Yawning Portal, as well as one for Icewind Dale: Rime of the Frostmaiden. .
"The DM might..."
These are not rules in the traditional sense. These are suggestions a DM might use to facilitate a subclass change in their game. Under "Training Time", we see (emphasis mine):
To change your subclass, your DM might require you to spend time devoted to the transition, [...]
The DM might also require an expenditure of money to pay for training, [...] This cost might be accompanied by a quest of some sort.
And under "Sudden Change", we see (emphasis mine):
Sometimes a character undergoes a dramatic transformation in their beliefs and abilities. When a character experiences a profound self-realization or faces an entity or a place of overwhelming power, beauty, or terror, the DM might allow an immediate subclass change.
These are not rules. These are suggestions to help a DM facilitate a subclass change, which is a pretty significant change. The idea here is to make the transition just as meaningful as where you came from and where you're going.
How would I personally work a subclass change?
I would ditch the limitation of "you can change your subclass when you would normally gain a new subclass feature."
The two methods offered give us distinct approaches for handling two different situations:
- A character says they want to switch subclasses some time in the future.
- A character says they want to switch subclasses very soon.
In either situation, I would work the subclass change at their next level up. In the first scenario, if they give me time to work in some plot, there is going to be some significant plot leading up to their next level up that will culminate in a change of subclass when they finally level up.
The second scenario is when I would use the "Sudden Change" guidance. If we are nearing a level up, and my player says "I want to change subclasses", I don't have time to work in significant plot build-up to the change, so I will opt for something like is suggested in this section at the next level up.
In either case, I am looking forward to the next level up, not the next subclass feature.
Waiting for a level up is how I would first express this to the player. If they felt that was too long to wait, I would of course see how we could work a transition even sooner.
What about rogues?
I present rogues as a use case for why I would break from the guidance given in the book. Rogue subclasses have a huge gap in features. Once a rogue gets their first subclass features at 3rd level, they don't get another until 9th level.
Say one of my players is a 5th level rogue and wants to change subclasses. I'm not waiting till 9th level to allow this. In this case, I would ignore the suggestion given in Tasha's Cauldron of Everything and still work the subclass change for their next level up.
Best Answer
Out of combat, this isn't even as good as the 1st level spell find familiar.
This sort of application of beast companions isn't all that creative. It's how wizards have been doing scouting for years with the find familiar spell. And honestly, find familiar comes with some features that make it superior for out of combat use. Here are some of the highlights from the spell description:
Beast Master Companions don't get any of this cool stuff, and this cool stuff is a day 1 feature for wizards.
This is the cost of doing business. Don't nerf your ranger's cool features.
In my experience, what you describe is pretty typical D&D stuff. I've never been in a game where this sort of thing didn't happen. We even have a question about it: How do I handle a player exploring the entire dungeon with his familiar? Much of the guidance there is helpful and relevant to your situation, but I would offer one particular quote from Kotrin's answer:
If this sort of thing is truly causing issues for you as a DM, talk to your ranger and work something out; but I would recommend rolling with it. Your ranger has taken this subclass for this feature. I'd recommend against nerfing it, especially since it's already not as good as a 1st level spell.