Are you able to apply your tool proficiency bonus when there is an overlap with a skill but you are not proficient in that skill?

Question

So for DnD 5e, XGtE expanded/clarified the use of Tool Proficiencies and very specifically talks about when skill and tool proficiencies overlap and what to do in those situations (advantage or added benefit).

My question is about the situation where you are proficient with a tool but not a skill and there is a potential overlap in an ability check, are you able to apply your tool proficiency bonus to that ability check?

For example, say you are proficient in "Smith's Tools" and a ability check comes up when examining a Sword. According to XGtE there is an overlap in skills with Arcana, History, and Investigation when examining metal objects. However you are not proficient in any of those 3 skills. How is that ability check handled?

1. Apply tool proficiency bonus to an Intelligence Check

or

2. Apply advantage to an Intelligence Check

Another example which is definitely nitpicking I will add is Mason's Tools. In XGtE they state there is an overlap in Perception when examining a structure crafted from brick. Now Perception from my understanding is what you are able to notice with your 5 senses rather than say a dedicated Investigation. So you wouldn't really bring out your tools (unless it is to enhance your senses like a spyglass but Mason's Tools only consist of a trowel, a hammer, a chisel, brushes, and a square).

So is XGtE suggesting when looking at a structure composed of brick you are able to do a Wisdom check and add your proficiency bonus (because you are proficient with Mason's Tools) w/o the need of pulling out Mason's Tools?

Answer 1

Without skill proficiency, the rules from XGtE don't apply.

The bonuses mentioned in XGtE only apply when you have proficiency in both the skill and the tool, and use of both the skill and the tool apply to the ability check:

Advantage. If the use of a tool and the use of a skill both apply to a check, and a character is proficient with the tool and the skill, consider allowing the character to make the check with advantage. [...]

Added Benefit. In addition, consider giving characters who have both a relevant skill and a relevant tool proficiency an added benefit on a successful check. [...]

If you lack proficiency in the skill, these rules do not apply.

Tool proficiency only applies when you are actually using the tools.

The rules for tool proficiency state:

Having proficiency with a tool allows you to apply your proficiency bonus to an ability check you make using that tool. For example, a character proficient with carpenter’s tools can apply his or her proficiency bonus to a Dexterity check to craft a wooden flute, an Intelligence check to craft a wooden secret door, or a Strength check to build a working trebuchet. However, the proficiency bonus wouldn’t apply to an ability check made to identify unsafe wooden construction or to discern the origin of a crafted item, since neither check requires tool use.

So in the example where you are examining a sword, you only get to add your proficiency bonus if you are taking the time to get your smith's tools out and use them. Cursory examinations would not benefit from proficiency with the tools.

In cases where a tool proficiency does apply, but there is crossover with a skill you are not proficient in, you add the proficiency bonus because of the tool proficiency per the first sentence of the quote above.