Can compressed sensing be used instead of intepolation for missing values?
Yes, at least in the above case it is possible.
Though it might not be computationally as cheap as other methods such as least squares based curve fitting methods.
I do not think injecting NaN gonna help, instead  let's look at the ignored data as dimension reduction feature of CS.
Here, the measurement matrix is basically an identity matrix that some of its row (120:130) have been removed:
A = eye(256);
A(120:130) = [];

Just be aware, for the case of measurement matrix you posed in your question, it might violate coherence requirement.
If we apply the above matrix, we cut away samples of 120 to 130. Now assume the resultant as measurement vector $y$ and $A$ as measurement matrix. This is how CS recovery algorithms work, e.g. in case of Fourier sparse signals, we have an incomplete set of measurements $y$ ,the goal is to find sparsest set $a$ such that if we apply A matrix, yields something close enough to $y$.
N = 2^8;
t=linspace(0,16,N);
x=sin(2*pi*t)+sin(2*pi*3*sqrt(2)*t+pi/4);
x=x';
Phi = eye(256);
Phi(10:17,:)=[];
y = Phi * x;
Psi=dctmtx(N)';  % Transposed DCT matrix is like inverted DCT
recovered=SolveOMP(Phi*Psi,y); % any solver available!
CS_intepolate=idct(recovered);
plot(CS_intepolate,'*'); hold on; plot(x,'r')


(blue line is CS interpolated and red is original signal)
BTW, as far as I understood, you suggest to mix the remaining data. See it as system of equations, blending the current equations into new ones, does not add any new information or constraint to the system, so it is not helping.
Also I believe there is connection with CS and data completion. You might find it interesting also.