Say what?
AFAIK, there is no mechanical shutter of any type sitting in front of and "protecting" the image sensor of any mirrorless camera. They use an electronic shutter, which essentially just "captures" what the exposued image sensor is seeing at the instant the "shutter release" button is pressed. It's not like DSLRs/SLRs, which have a mirror redirecting the image through their optical viewfinder (which has to flip up out of the way during exposure) and behind that a mechanical shutter that controls the exposure of the image sensor hidden behind it. Someone might have designed a mirrorless with a shutter that drops in front of the image sensor, purely to protect it... but I'm not aware of any that use that.
In fact, when a mirrorless camera is on and it's electronic viewfinder is active, the camera's image sensor is what's capturing the image being displayed on the tiny LCD in the viewfinder.
The lack of a mirror (hence, "mirrorless") AND mechanical shutter are what allow mirrorless cameras to be smaller... to use a much shorter lens flange to sensor distance than DLSRs require. The electronic shutter also makes it possible for them to operate nearly silently, since there are no moving parts. Those are all good things.
The down side of this is that.... 1. the image sensor is on all the time whenever the electronic viewfinder is in use (much like a DSLR in Live View mode, but not when using its optical viewfinder), drawing power from the battery, gaining heat, and putting a lot more accumulated time on the image sensor and related electronics... and, 2. compared to a DSLR, the image sensor in a mirrorless camera is much more exposed to dust and/or possible damage during lens changes. Hence, yes, a mirrorless camera's sensor is less protected from exposure to the sun, too.... careless handling might damage the sensor, especially if the sun's heat is concentrated onto the sensor by a lens (ever used a magnifying glass to set things on fire?)
That's the sensor you're looking at inside that Sony A7III mirrorless, when it's lens is removed.
I"m not sure why it's relevant, since the question is about mirrorless... But Canon hasn't made a rangefinder camera using a cloth shutter in over 50 years. I think the Canon 7s, discontinued in 1967, was the last RF they made with one. Some of their SLRs continued using a cloth shutter a bit longer. The A-series (AE-1, A-1, AT-1, AV-1, etc.) all use a cloth shutter. The last of those (AE-1 Program and AL-1) were discontinued over 35 years ago, in 1982. Some Canon models concurrent with those (such as the F-1 in three versions) and all Canon SLRs and DSLRs since that I'm aware of have used a vertically traveling, metal bladed shutter (Copal or a variation of it). In the later cameras these are electronically controlled for accuracy (but so were some of their cloth shutters, too).
Yes, the shutter (cloth or metal) of a DSLR/SLR also can potentially be damaged by the sun too. Except in DSLRs and SLRs the shutter is hidden and protected behind a mirror most of the time.... except for the very instant of exposure when it flips up out of the way. Most of the time the mirror is in position to redirect all or most of the light through the pentaprism and viewfinder. (Some DSLR/SLR use a semi-transparent mirror to allow part of the light to pass through to metering and/or autofocus sensors behind.) It's different with rangefinder cameras. They don't provide "through the lens" viewing and have no mirror (allowing them to be smaller and use a shorter lens register, like mirrorless digital). A RF's shutter is more directly exposed to light coming through the lens or whenever the lens is removed for the camera.