Can heavy tires handle anything that the trail can offer? I got a Slash and it came with Bontager XR4 Ellite tires.

The same evening at the park I had a puncture. Turned out the offender was a thorn to the sidewall (I rode through some cut off and now dry bushes and tree stems).

I intend to ride the bike over snow, ice, mud, sharp rocks, large round rocks, road, but mostly trails, that can sometimes have sharp stones or thorny plants. And wound NOT want to patch both my tubes every time I go through some strawberries.

So I am considering moving to very heavy tires, e.g. Scwalbe's Big Betty or Wicked Will. Would this mitigate my concerns?

Alternatively, the current tires and rims are marketed as "Tubeless ready". Maybe it would be a better bet to move to tubeless setup? But then, why didn't they sell the bike in TL configuration, if that would be better?

I do not ride at competitions. My goals are to be able to ride to the top of the mountain without lagging behind my friends, and then enjoy the descent.
I certainly feel your pain about having to repair tires all the time. I've been stuck 10 miles from the trail head with a cut in a tire that caused me to constantly stop and add air as I limped back (second tube of the day).

Bigger, beefier tires will help, but if your plan is to make it up the hill with your friends and not be totally gassed then I wouldn't recommend full DH tires. I switched out my rear on my AM rig from a 2.35 to a 2.5 and boy does it chug when laying down the power. On the other hand, I can point it downhill and not even worry about it, and the grip is fantastic.

For good, all-around trail tires check out something in the 2.35-2.4 range that's made for All-Mountain/Enduro as these usually have thicker sidewalls and harder rubbers.

As for going tubeless, that will certainly help. The sealant will handle most of the punctures that you're likely to get, but I have heard of folks getting numerous punctures that have slowly bled the air from a tire. The problem with tubeless when riding more aggressively and on harder terrain is burping, where the tire briefly comes out of contact with the rim and you lose some air pressure. No amount of sealant can prevent that.

My bottom line recommendation is to go with a split tube/tubeless setup. You can hit everything from fast XC trail to full on DH and be covered in the case of punctures and big hits. Sure it's a little heavier, but you won't be running big huge tires either.