In these three Legend Lore articles Mike Mearls talks about the fact a major design goal of D&D 5e is to Unite the editions.Specifically allowing the core game to be modified to play similarly to one of the past editions of D&D.
Uniting the Editions
While not all mechanics were carried forward from past editions, vancian magic was one of them. Vancian magic in a modified form with at-will cantrips, prepared spells, and rituals. The most direct answer to your question magic is what it is because that how it was presented in OD&D, AD&D 1st, AD&D 2nd, and D&D 3rd. It was modified in light of the experience with D&D 3e and D&D 4e and for reasons outlined in this article.
I could leave the answer like this but I feel it not complete. Some will wonder why Vancian magic in the first place?
It started, like in many mechanics in OD&D, with Chainmail. By the 2nd edition of Chainmail, wizards of varying power were introduced in the fantasy supplement. The four levels were Magician, Warlock, Sorceror, and the most powerful the Wizard. The difference between the different levels was not only in the power of their spells but the number of times per day they could cast spells.
When Gygax developed his Greyhawk Campaign he decided not to use Dave Arneson's system of magic reagents but rather was inspired by Jack Vance's Dying Earth series to create the familiar mechanics of spells in a spell book and the magic-users memorizing a limited selection of spells from the book.
This is corroborated in both Jon Peterson's Playing at the World and Kent David Kelly's Hawk & Moor series.
The mechanics are designed in 5e to reflect the spell memorization of classic editions of D&D which were inspired by the literature that Gary Gygax read most importantly Jack Vance's Dying Earth which were adapted from the Fantasy supplement of Chainmail which was used by Dave Arneson in his Blackmoor campaign.
Average The Skills
If he has to use two skills, average the two skills together and then make one roll. In this case, that'd be a single roll to get 50 or below, since he has 50 in both skills (so the average is 50).
If he was better at one skill than another, it'd look slightly different. Say he has a 50 in Stonecarving and 25 in Artistry. That makes the average of them 37.5, so he'd have to get a 37 or below (or a 38, depending on how you want to round).
That basically treats it like he's using both skills and has to succeed on using them in combination, rather than having to succeed on separate rolls for both. It also keeps it to a single roll with similar odds, and is relatively simple to implement for players.
Alternative - Geometric Mean
The downside to averages is that if you're really good at one skill (say 100 in Stonecarving) and really bad at the other (0 in Artistry), you still have a 50 in the combined skill. That might not be what you had in mind, as someone with no artistic talent doesn't suddenly gain it just because they are working with stone.
In this case, an alternative method is to take the Geometric Mean. For two skills, that is this formula:
$$\sqrt{skill_1 \cdot skill_2}$$
So, if you have 100 in Stonecarving and 0 in Artistry, you do \$100 \cdot 0\$, which is 0. The square root of that is 0. As a result, you now need to at least have 1 skill point in Artistry in order to attempt the combined result. If you did have Artistry 1, you'd get \$100 \cdot 1 = 100\$, the square root of which is 10. As you add points in Artistry, your chances will quickly increase.
For my previous example of 50 and 25, you'd get \$50 \cdot 25 = 1250\$, the square root of which is 35.3.
The main downside to this method is that in a tabletop game, it's extremely hard to calculate without a calculator. Even with one, it requires a more complicated understanding of math and is more time consuming. If you put this in a rule book, there will be people who won't understand what you want them to do. For something like a video game where it's calculated by the software, that isn't a problem.
(Thanks to Peteris and Vatine for the suggestion!)
Alternative - Minimum/Maximum
A very simple method for combining skills is to use either the minimum skill in the two of them, or the maximum skill in the two of them. The maximum means you're just using the skill you're better at, while the minimum means you're using the skill you're worse at.
In the case of the minimum, it simulates the idea that you have to succeed on what you're weaker at in order to accomplish the goal. This lets you do it in a single roll, and is very easy to understand. It also has some issues, in that if you're extremely good at Stonecarving and so so at Artistry, your Stonecarving gets ignored in this system as you only roll on your lower one (Artistry).
Because of that, I don't think it really accomplishes what you intend very well, but it's ease of use is a significant upside over the other suggestions.
(Thanks to Neil Slater and Ellesedil for suggesting.)
Best Answer
In common practice a d100 is effectively a 0-99 roll, with the stipulation that 0 be treated as 100
The game needs a way to roll 1 to 100 with equal chances, and no chance of getting zero.
Let's start by just looking at how we are set up to roll the results from 1 to 99, and then we'll get to the special case of getting 100.
In order to have a practical way to get the numbers 1-99, we roll a double-digit and a single-digit d10 together, and add them arithmetically. We add the two numbers we literally see, so a "90"and a "3" is 93; a "00" and a "3" is just 3; a "90" and a "0" is just 90, etc.
Two problems initially: we have no way to get 100, and we have the possibility of getting a total zero by rolling "00" and "0".
Solution to both: count the "00" and "0" combo as 100, instead of zero. With this simple adjustment, we have exactly what we want: a 1-in-100 chance of getting all possible results from 1 to 100, and never getting a final zero.
Admittedly, it is a matter of convention
Understandably, you normally treat a "0" on a regular d10 as "10", and you could continue to do so even when using it in d100, such that if you rolled "80" and "0" and treat the "0" as 10, then you get 90. But game communities tend to converge on one common practice or another, and the one outlined above is the one that has taken hold. But technically, you could use either method, as long as everyone at the table understands and agrees.
A final historical note
In the old days, few players had two d10s to enable rolling a d100 all in one go (maybe because dice sets were more expensive relative to income back in the day). We'd roll our single d10 for the tens-place, and then pick it up and roll it again for the ones-place. This made the zero-substitution rule very exciting and suspenseful! If my initial one-die roll was zero I'd be thinking, "Dang, I probably will end up with just a 1-9, but I've got a shot at a 100!". And everyone around the table would be thinking the same thing, and would watch with tense anticipation what the second roll was going to be.
This dynamic added some fun to the game (that you don't "feel" when rolling two d10s), and might help explain why the method "stuck."