Here is my take on this:

Looking at FRED's data, TFP growth in the US has averaged 0.7%/year with an annual standard deviation of around 1.1%/year since 1955. The data looks consistent with both a linear trend and an exponential trend, essentially because the time window is too short, so I think based on this data alone it's impossible to get a large likelihood factor for one model vs another in either direction. My estimate of the value added of increasingly elaborate statistical methods on that evidence is low.

The "fast growing economies" data the paper looks at is highly misleading because it's a well-known fact that growth in these countries slows down as they become less poor compared to the frontier economies. Therefore we'd expect all growth in these countries to slow down as they become richer, not just TFP growth. It's not unexpected that a linear model can get a better fit in this particular case. The really interesting question is whether this is also true of frontier economies, i.e. ones that can't "catch up" to countries that are richer and more productive than they are.

The paper also looks at data from before 1955, but here as you say it needs to start introducing new pieces into the piecewise linear model, which undercuts the persuasiveness of its argument considerably. The main problem is that a piecewise linear function with only two pieces already has four free parameters, compared to an exponential fit which has only two. It's not at all surprising that a model with four free parameters fits the data much better. Adding more breaks makes this problem even worse, since as far as I can see the paper never puts the exponential growth model on parameter parity with the piecewise linear growth model for a fair comparison.

My guess is that if we look at the post-1800 (or post-1600) period, the exponential growth model will outperform any linear growth model at parameter parity. This means that a constant exponential growth rate model beats a constant linear growth rate model, and if you complicate the linear model by adding time-varying growth rates with punctuated equilibria whose arrival times are sampled from a Poisson process or whatever, there will be a similar way to adjust the exponential growth model with extra parameters via autocorrelations etc. such that it will still outperform the linear model if they are working at parameter parity. Of course in the "too many parameters" regime the linear model will be able to approximate the exponential one sufficiently well, so in this case I think the performance could end up being comparable in sample but the exponential model will not underperform. I'm willing to bet real money on this.

There's also a broader point here that TFP itself is a dubious measure. It's a residual estimate of a log-linear regression of GDP on labor and capital stocks in what is certainly a misspecified model (Cobb-Douglas, with only labor and capital inputs!) There's also a well known phenomenon that if in your sample labor and capital shares of national income have remained relatively stable, you'll end up estimating that a Cobb-Douglas function is a good fit for it simply because a Cobb-Douglas is defined by the property that the factor shares of income are constant. I think taking it too seriously is a mistake and the fact that such an artificial measure may have sub-exponential growth definitely doesn't imply that "growth is linear and not exponential".

Overall I think the paper is not informative or interesting, and seems to be motivated more out of a desire to be contrarian than anything else.

Update: I've dug into the data a little bit more and it seems like the main advantage of a linear model on near-term data is that it explains away the "great moderation" puzzle. The paper does point this out, but because it doesn't do any log likelihood comparisons it's not easy to appreciate how crucial this is. An exponential growth model with additive noise is only somewhat worse on the data since 1955, perhaps a likelihood factor of ~ 2 favoring the linear model. However, an exponential growth model with multiplicative noise is overwhelmingly worse, with likelihood factors of 50 or more in favor of the linear model.

This is a subtlety that I think the paper doesn't emphasize enough. It does note that the linear model has no heteroskedasticity puzzle, but in fact separating the noise structure from the trend structure takes away most of the edge the linear model has over the exponential model on postwar data. Likewise, adding noise decay to both the linear and the exponential model gets their likelihood ratios in the same ballpark, with a likelihood ratio of < 2 favoring the linear model.

For the moment I've updated towards recent data providing somewhat stronger evidence in favor of a linear model than I'd thought, but I think extrapolating the low noise predictions of the linear model into the future is dangerous practice if you want to generate forecasts & probably leads to overconfidence. I think if the paper had actually mentioned this point about log likelihoods I would have been more sympathetic. As a result, I've crossed out the last sentence of my original comment.

When you zoom in on an exponential curve close enough, it looks linear

Stepping back a bit, it seems to me that the big puzzle of growth theory is that we see super-exponential growth over the very long term, but sub-exponential growth over the last several decades. I don’t think we yet have a unified model that explains both periods.

I thought that was due to population growth stopping?