Yes, any well must be able to be shut in. I phrased that poorly. What I meant was that not all wells respond to a shut in in the same way. It's not uncommon for things to take months to recover to the same production levels after a shut in. (At least for conventional wells -- unconventionals are significantly different.)
(Major caveat here: I'm an exploration geologist who later went into remote sensing, so my knowledge of actual operations is pretty spotty. Feel free to ignore me on this, but I'm going to keep rambling anyway.)
I was referring to a lot of onshore wells in declining or very small conventional fields. In many cases, a shut in that lasts for more than a few weeks basically leads to a P&A, as production won't recover without a workover, and a workover isn't worth it. That's what I meant by "not designed for it" -- production is expected to be sub-par after a shut in and a workover was never in the plans. A shut in means a death sentence for some wells, so operators prefer to keep them trickling along when possible, as it leads to a higher EUR. (Also most of these are hooked up to stock tanks, not production gathering pipelines, so it's a bit of a different world.)
However, I'm extrapolating from the very little experience I've had with that. Literally one tiny onshore field in TN (yes, Tennessee) that I worked in/with 15 years ago a summer after undergrad (I was actually mostly collecting seismic with a tiny source and a short string of geophones from the back of a pickup truck, so not terribly closely related). I may very well be way off base.