Do photons take all paths or not?

It is a bane of small scale physics (e.g. quantum mechanics) that we have absolutely no natural intuition. By natural intuition I mean something along the lines of the following. Take this question "if a ball falls down under gravity after a meter of falling does its speed exceed 100km/h". You know the answer is no because you have seen balls falling down from tables. We develop a theory (a model) that answers this question quantitatively (Newtonian mechanics). At the end we are very happy because our natural intuition has matched with the quantitative answer. Schematically it looks like

$$ \text{Intuition} \dashrightarrow \text{Phenomena} \to \text{Model} \to \text{Quantitative Predictions} \to \text{Intuition} \checkmark$$

Let's take another question "which slit did photon go through in a double slit experiment?" It is not so obvious anymore because no one has seen photons go through very tiny slits. We don't even know whether this is a meaningful question. Ignoring any natural intuition we jump directly to developing a quantitative model. We can tell the intensity distribution of photons because we can detect them and count them and our model agrees with this distribution.

However, we still haven't answer the original question. We now try to interpret the theory and come up with intuition about this phenomena. Again no one has "seen" photons. We can talk about photons behaving as waves (usual interpretation Quantum mechanics) or we can talk about photons taking all paths (path integral interpretation). Note how classical these interpretations are because we only have natural intuition for classical physics. The quantitative thing (i.e. the model or if you want the math) is undisputed. However, what that math "means" is sometimes unclear, precisely because of the lack of any natural intuition. Schematically the analogous diagram looks like:

$$ \text{Phenomena} \to \text{Model} \to \text{Quantitative Predictions} \dashrightarrow \text{Intuition ??} $$

Photons are not little balls of classical matter. In particular they have not a definite trajectory nor a position. Their description needs a suitable notion of quantum state in a suitable Hilbert space.

In some, very special, regimes states of single photons can be approximatively described as particles moving along straight paths (para axial states). Also several paths simultaneously according to corresponding probabilities of a certain path.

Conversely, states of a very large number of photons (coherent states) can be described by classical waves to some extent.

Each such description is quite partial and it cannot capture all facets of photon phenomenology which is fully encompassed by the complete quantum mechanical description in the Hilbert space.

The folklore picture where a photon runs all possible paths is actually a popular illustration of the Feynman path integral method to deal with quantum particles. Actually it is a quite technical machinery which cannot be reduced to this popular representation. Literally taken it may produce mistakes.

Do photons take all paths or not?

Yes, they take all paths. This can be seen by single photon sources and:

Double slits Diffraction gratings Lenses Etc.

For me, the diffraction gratings are the most convincing.

the photon travels in a straight line,

This is clearly not correct in a myriad of experiments. Particularly where there is diffraction.