My MIL’s dog has constant health problems and is very old. Why doesn’t she just have him put to sleep?

Depends on whether he's in pain or not. But how is this YOUR business?

Yes it would.  But your MIL is the one that has to make that decision.  As long as she has the money to pay for vet bills, then its up to her when she puts him down.

That is for your MIL and her vet to decide, not you

The dog is dying. Lethargic & no appetite means heart failure. The dog is suffering. It only has weeks to go a painful death.

Yes it would..............

This is a very difficult decision to make. Really it's up to your MiL and the Vet she uses to come to a decision

I happen to believe that as a dog gets older, it's all about quality of life.   If your MIL's dog has a pinched nerve and so 'constant pain', then he's unlikely to be enjoying life any longer.    BUT he's your MILs dog so it's going to be her decision to make even if you could have a quiet word re her dog's 'quality of life'.   It's a hard decision for anybody to make.

ps   When one of mine, with various age-related problems, stops eating, then I KNOW it's time to give peace - we never allowed 'lingering'.

If the dog is not in pain, why kill him.  Would you like to be killed just because you are old and have a few health issues?  I realize you said "constant pain" but, is the dog whimpering or just moving slowly and you "think" it is in pain.  Watch the dog for a while and if you really think it is in terrible pain, then you can always suggest that putting it down might be the best thing to do.  Being a pet owner myself, we don't want to let our pets go.  We think we have a moral duty to make the pet as comfortable as possible and let nature take its course.  Unless it is in agony, I would not put down a pet.  Not anymore than a human being.

she must be very attached to it.if she wants to pay for vet bills then its up to her

I think so. I don't own the dog. Nor do you. Keep your peace.