"Non-stationarity" is really an umbrella term; the truth is, there is no single authoritative model of asset returns, and really there will never be one. This should not preclude all statistical analysis though, and it is done by making simplifying assumptions, just like in every other case, and in every other field. Your claim is that they are too strong in this case, but it's a claim that can be fairly easily shown empirically or in simulations, and it really should be IMO, particularly since other bootstrap methods exist.
I agree about algorithms; rather unfortunately, this is true in more than this one paper. There certainly is movement towards requiring people to make their code fully available, but we are not quite there yet. But if you describe your failure to replicate, this is definitely a strong argument that the authors would IMO need to address.
Ignoring transaction costs would be a major problem if the paper's main point was "we found a strategy returning X% above market, it's awesome and people should give us money" -- but this is written for a very different purpose and audience. That being said, today it would not be published without a transaction cost analysis -- but I would have no problem with them saying "with such-and-such costs, profits are not there any more", it would not invalidate the paper at all. But at the time it was written, TC analysis was not as standard in academic literature as it is now.
I agree with you about TA, and TBH most serious researchers are of the same opinion, and have been for a long time -- even at the time of publication, it was a bit of an outlier, and this is not a particularly popular area of research (how many of those citations are in recent top journal articles?). Forex was a bit of an open question last time I checked, but it's been a few years, not sure if it still is.
