That's the right approach. But, as the other commenter pointed out, there are a number of complications which mostly center on the inability to distinguish cancer from self. Many of the mutations that make cells cancerous are not expressed at the cell surface, and the only way to probe the interior is through MHC molecules which act as little windows into the cell. More specifically, during normal metabolism the cell will chop up the proteins its making and display them on the surface for immune surveillance. The immune system can use these because its been trained against EVERYTHING SELF (in the thymus) so when something goes wrong in a cell and surface expression looks different it can usually recognize this. Of course by the time cancer has developed something has gone wrong in this process normal process in any number of ways.

Even if the challenge was just "recognize MHC-presented cancer antigen" in therapy you have to a) determine what section of the mutated protein would be displayed in MHC, b) determine and develop an antibody (or antibody-like molecule) how to recognize specifically that (and not the non-mutated version which will be present in healthy cells), then c) either genetically engineer the patient's immune cells to recognize it, or hijack the recognition process in some other manner.

Each of these steps is very very difficult and we're only just developing the computational and experimental tools to do these for any patient, let alone every cancer patient. I won't go into it, but you also have to think about cancer as a living organism susceptible to evolution so if you don't hit hard and fast to wipe it out all at once, you select for mutations that evade your treatment. This is made especially challenging considering that cancer usually has some mechanism gone awry that leads to increased growth and mutation rates so that they're even MORE likely to evade your treatments then a generic cell - think of it kind of like antibiotic resistance in bacteria