I would suggest that it's best to think of it as the exact opposite, mammals came from ectothermic ancestors, and gradually began to develop what could be considered "extreme hyper-metabolic states" in comparison, which offered various advantages (denatures the proteins in many microorganisms, allows for consistent physical performance and activity at night when when most of the reptile predators were inactive) at the cost of needing a much higher and constant food/energy intake, and began to spend most of their time in such a state until it became "normal". However many mammals still require the ability to enter significantly lower metabolic states to survive every winter.

A good example is Golden Mantle Ground Squirrels for example, which need to consume enough omega-6 PUFAs (I believe linoleic acid is the primary one involved here, not sure about the others) in order to activate a switch in metabolism that ultimately drops energy use and puts them in a feedback loop to keep them there. [0] They'll periodically have rises in metabolism followed by some activity, and then drop back down into torpor throughout the winter. Squirrels that fail to kick off this change strongly enough are at big risk of burning too much energy and failing to survive the winter. Far less extreme and more well known are bears who also undergo significant fatty acid content changes. [1] Of course, metabolism is hideously complicated, and is effected by things like sunlight (touches vitamin D related pathways, among other things), vitamins (B vitamins are particularly notable) and fatty acids, proteins and carbs which all have have their own individual effects which can change in magnitude or even result in the reverse of what they "should" be doing. Look at this fucking shit and despair of ever trying to make a single correct context independent statement.[2] A perfect example I just found while looking up the other study is that while golden mantle ground squirrels appear to require linoleic acid to enter torpor, this study found that removing linoleic acid 19 days prior to hibernation enhanced how deeply they entered torpor. This isn't surprising to me, because I've heard of the possibility of linoleic acid being potentially exothermic in unusual circumstances, like extreme amounts. So laying off it once the cycle has started may allow the various feedback loops in place when monounsaturated fats run the show to more strongly express [3]

I personally consider all mammals as retaining echos of our cold blooded heritage and capable of enter lower metabolic states, both daily and seasonally (to a widely varying degree) depending on the aforementioned very complex set of conditions. This study seems to agree with me. [4]
So I also personally find it quite plausible that if evolution begins selecting against the hypermetabolism that was adopted, then the genetics are still ready and waiting for re-emergence, especially on an island where disease pressure is going to be much more limited. Evolution seems to like adding and repurposing things we already have, adding layers upon layers and rarely ever actually removing things.

As for why lizards wouldn't dominate the island, there were and still are lizards present, unlike the goats who didn't manage to evolve back into lizards fast enough to escape notice by humans and thus extinction (though a quick search seems to show that many lizards on the island are also considered endangered). That said lizards are generally predators/omnivores, whereas the goats are herbivores, so they weren't likely to be strongly competing for the same niche.

[0] "The Influence of Dietary Fatty Acids on Hibernation by Golden-Mantled Ground Squirrels (Spermophilus lateralis)": https://www.journals.uchicago.edu/doi/10.1086/physzool.65.5....

[1] "Lipidomics Reveals Seasonal Shifts in a Large-Bodied Hibernator, the Brown Bear": https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6474398/

[2] Metabolic pathway charts:

- The famous Roche chart: http://biochemical-pathways.com/#/map/1

- Hand assembled map of the above https://faculty.cc.gatech.edu/~turk/bio_sim/articles/metabol...

[3] Short-Term Variations in Diet Fatty Acid Composition and Torpor by Ground Squirrels: https://www.jstor.org/stable/1383506

[4] "Daily torpor and hibernation in birds and mammals": https://onlinelibrary.wiley.com/doi/10.1111/brv.12137