But that's you making extra room for it. Most officialy published material would call it an investigation check and call it a day.
I'm not defending it but it inherritly lowers the value of tools.
No, all officially published material would call it an Intelligence (Investigation) check. So what it
should look like is:
“Make an Intelligence check, plus proficiency if you’re trained in Investigation.”
“I’m not, but my Smith’s Tools proficiency seems pretty relavent here. Can I add that instead?”
“Sure.”
Of course, this exchange is clunkier than the previous one, which is why I just call for the ability and let the player suggest a proficiency instead of offering one off the bat.
Now, I’m aware that despite both what the rules say and how published adventures are presented, many DMs just ask for skill checks like we’re still playing 3e. Getting a tool proficiency to apply with one of these DMs can take a bit more coaxing, but it generally looks something like this:
“Make an Investigation check.”
“With Intelligence, I assume?”
“Huh?”
“Am I adding my Investigation proficiency to my Intelligence modifier, or another ability?”
“Oh. Yeah, just a standard Investigation check, so Intelligence plus Investigation.”
“Ok. I’m not trained in Investigation, but it seems like my knowledge of Smithing would be pretty relevant here. Can I add my proficiency bonus for Smith’s Tools to my Intelligence check instead?”
“Hmm... I guess... But you’ll have disadvantage!”
“Wha... Ok, I’ll just roll straight Intelligence, I guess.”
That’s a horribly clunky exchange and might not actually lead to getting to add your tool proficiency, but that’s on the DM for trying to run 5e like another edition.
Then there are the DMs who let players initiate rolls. They’re the easiest to get tool proficiencies to work with.
“Hey, can I make an Intelligence check with my Smith’s Tools proficiency to (whatever)?”
“You can certainly try.”