There are a number of exceptions: hotel restaurants, bartenders, minors, students
That is incorrect all of these employees are subject to the Fair Labor Standard act. Everything previously stated applies.
business not covered by the Fair Labor Standards Act with gross annual sales of $110,000 or less
So that's an interesting one, because Montana is even stricter than the what is laid out as an exemption federally. Which is a buisnesses that makes less than 500,000. But those buisnesses also cannot engage in Interstate Commerce per federal law. In the modern world it is nearly impossible to have a buisness making any amount money that does not engage in some form of Interstate Commerce. So functionally that exemption is meaningless.
You did skip a few types of employees that are actually exempt like movie theater employees. Originally they were exempted from overtime and federal minimum wage provision of the FLSA. However, this has since been changed and now they are only exempt from overtime regulations.
Now to your last part which is the crux of your misunderstanding
Some states set subminimum rates for minors and/or students or exempt them from coverage or have a training wage for new hires. ...Such differential provisions are not displayed in this table.
This only applies to the wage they receive before tips. Still per federal law they must make at least $7.25 an hour after tips.
What this paragraph is saying is that a state may have a minimum tipped wage of $4.00 per hour, but a provision for students that sets it to $3.00 per hour. The table does not reflect this.
An individual state law cannot supersede federal law. So these employees still must make a minimum of $7.25 an hour. That is base tipped minimum wage plus tips. If the minimum wage, be it 3 or 4 dollars to use the example, and the person's tips add up to less than 7.25 an hour then the buisnesses must cover the difference. Unless the buisness is exempt, which is functionally impossible for a restaurant.
This is all to say yes, all servers do indeed make at least the federal minimum wage of $7.25 an hour. If they are not making at least that they should contact the department of labor for unpaid wages.