First, the degrees being too narrow is a non-starter. The stick he uses for the 22 degree markers provides the accuracy, and a wider circle would compensate for the smaller, 13 degree sections. And the Pathfinder Lander (renamed the Carl Sagan Memorial Station) IMP (Imager for Mars Pathfinder) would absolutely be able to accurately match any arbitrary degree of rotation. A quality Servo can have a position accuracy in the sub-degree range!
That said, the main reason that Mark uses Hexadecimal is that he needs a way of communicating with NASA that someone will know, without having the luxury of being able to communicate with them how. It's a Chicken or Egg problem. There is no standard base-10 language set. But Hexadecimal is used throughout computers everywhere, and by luck one of the other astronauts had a table on their laptop that Mark could use. Without that table, he would have had to create something from thin air, and hope that they could reply back in a reasonable manner. The ASCII to Hexadecimal was an already established protocol that they could both easily use without having to make something up from scratch, and the Table provided it.
Mark to Camera: I figured one of you guys kept an ASCII table lying around. And I was right.
Ladies and gentlemen, I give you super-nerd Beth Johanssen.
Followed by a nameless Engineer immediately understanding it.
Engineer: I know where he's going with this.
Any nonsense about not being able to send numbers or punctuation using a decimal notation is ridiculous, as both numbers and punctuation can be spelled out, and Mark and NASA engineers could work out short hand to prevent needing to spell out longer words.
Of course the Movie Portrayal is fairly non-realistic. Why would any engineer bother with a relatively hard to read multiplication table instead of the much more common linear table, which tend to have Hex, Octo, Decimal, and AlphaNumeric side by side. Why bother with a Question Mark when Hexadecimal has a Question Mark in it (0x63) AND they don't even bother to use punctuation when they send him anything. They also didn't use a space, leaving Mark to figure out what they mean by context.