Girma, Knight et al do it right.
They DO look over a much longer time; look at figure 2 of the paper. That’s what needs to be done if you are testing ideas about effects with a long period. Just eyeballing 150 years of instrument record can only give weak support to the hypothesis; the look at longer periods of time is essential.
Note also that their model is “quasi-periodic”. It is not a simple sine wave with a definite period. It is rather shows a characteristic time scale for changes, but shifts up and down somewhat chaotically at that scale. That’s pretty standard for these kinds of effect.
Finally, although you’ve agreed that the long term underlying linear line is unrealistic, you use it crucially for “predicting” or “falsifying” your supposed model into the future. You need to look at tools for identifying a periodic (or quasiperiodic) signal on top of a base trend that is NOT linear; because there’s a heck of a lot more going on with climate that you can capture on such scales with one line a sine wave. There ARE such tools, but as I’ve said, you really need a professional statistician to deal with that. It’s not trivial. I just work at the level of basic significance tests for regression lines and so on, which are okay as a ball park starting point but not really up to a proper hypothesis test.
I doubt any professional statistician would be much interested in how you’ve made your proposal, especially as there is no physical basis whatsoever being proposed which could be the basis of a test of prediction against data.