Does the frequency response \$H(j\omega) \$ exist for system with unstable poles?
Say we have a system \$F(s)=1/(2−s)\$ with a pole at \$s=2\$. In the time domain this function is \$f(t)=−e^{2t}\$.

No. The sentence started with system, and suddenly switched to function. To get the (time domain) function, an input was assumed to be applied to the system; viz, the impulse input.
What is done next, is the Laplace transform of the impulse response (and not the system). Of course, for this unstable system, the impulse response diverges faster than any signal represented on the imaginary axis (pure sinusoids) will converge. So the Laplace transform doesn't exist. From this, you cannot conclude that the evaluation of \$H\$ at \$s=j\omega\$ is meaningless.
Even for unstable system, the response to an input has two parts; a natural response (zero input response) and a forced response (zero state response). Evaluation of \$H\$ at \$s=j\omega\$ still represents the forced response part IIRC.
A confirmatory simulation
sys = tf([1], [1, -0.1]);

time = [0:0.001:5]'; u = sin(2*pi*10*time);

% evaluate expected forced response at w=2*pi*10 rad/s
[m, p] = bode(sys, 2*pi*10)
% m =  0.015915

% simulate
% since our input sinusoid isn't
% really starting from t = -infinity
% give an adhoc initial condition
% to prevent the natural response from
% running away.
lsim (sys, u, time, [-.016]);

% observe the actual response
ylim([-0.02, 0.02]);