Let’s not be economical with the truth here. Cars don’t go limp home mode because the oil doesn’t get hot enough, (otherwise all my retired clients would be having that problem) . A failed to open thermostat is likely to go into limp mode or check engine light on. Not sure what motors you are talking about but to the best of my knowledge only the diesels (sold and serviced Mazdas up till 2 years ago) were prone to giving a check engine light if they weren’t given a good long drive but mainly because the DPF tended to block up.