(a) "the total magnetic "effect" (as in Ampere's law) should be B times the length of the bold line" No: the Ampère's law line integral is of the field component along the chosen line, but in your case, $\mathbf B$ is at right angles to the bold line. Setting this aside, I'm afraid that your argument, starting with field strengths at different points along a line, doesn't have a sound basis and isn't valid. [See, for example, nasu's comment, "magnetic effect is not a standard quantity. How do you define it?"] That's why the prediction of an infinite field at the centre of the semicircle is wrong.
(b) For the straight wire the field at a point (say on your bold line) is not due just to the element of wire nearest that point, but due to the whole of the wire (though points further away contribute less, in accordance with the Biot-Savart law). The centre of an arc is not the only place where fields from different elements of a wire re-inforce.
(c) Bending the wire into a semicircle (of radius $R$) will increase the field somewhat at the centre, O, of the semicircle, compared with its value at the same distance, $R$, from the centre of a straight wire of the same length ($\pi R$). This is because no element of the wire will be further than $R$ from O, and because $\sin\theta$ in the B–S law will be 1 for all elements of the semicircle. It is a good exercise to evaluate the field in the two cases, using the Biot–Savart law. [I find that for the semicircle, $B=\frac{\mu_0I}{4R}$, whereas for the straight wire $B=\frac{\mu_0I}{7.45\ R}$.]
