[RPG] Does the Rogue’s Reliable Talent feature work for thieves’ tools, since the rogue is proficient in them

As of 2019, the new Sage Advice Compendium clearly states that they don't interact.

These aren't allowed to work together on a single check.

Can the rogue’s Reliable Talent feature be used in conjunction with Remarkable Athlete or Jack of All Trades?

No. Each of these features has a precondition for its use; Reliable Talent activates when you make an ability check that uses your proficiency bonus, whereas the other two features activate when you make an ability check that doesn’t use your proficiency bonus. In other words, a check that qualifies for Reliable Talent doesn’t qualify for Remarkable Athlete or Jack of All Trades. And Remarkable Athlete and Jack of All Trades don’t work with each other, since you can add your proficiency bonus, or any portion thereof, only once to a roll.

An earlier Jeremy Crawford tweet restates this:

Making sure this is clear: these features work together on a character. They don't work together on the same ability check.

This is definitive, and contradicts earlier apparent rulings, i.e. from the 2017 SA compendium.

It's not explicitly specified which ability you would use to employ these tools. Ultimately, it's up to your DM which ability is used for any ability check, and the default matching of skills to abilities is only a default. Both the Player's Basic Rules and the PHB include this section in Chapter 7:

Variant: Skills with Different Abilities

So it is up to your DM to decide which ability or skill applies to any ability check you make. With tool-based ability checks, this is even more the case, since they don't have the defaults that skills come with.

However, the 2 tool proficiencies in the question are both included in the list of examples of potential Intelligence checks.

Other Intelligence Checks.

The DM might call for an Intelligence check when you try to accomplish tasks like the following:

• Communicate with a creature without using words
• Estimate the value of a precious item
• Pull together a disguise to pass as a city guard
• Forge a document
• Recall lore about a craft or trade
• Win a game of skill

So the developers must have thought using these tools would generally be Intelligence checks.