Water – Plumber told me it’s unsafe to drain the water heater — what should I do

This is not a good example as the venting pipes are right below some windows which can leak exhaust gasses back inside the residence. This would be illegal in the state of Michigan.

I realize this is an old question, but it's hot once again and the question may come up for someone else.. so I thought I'd venture an answer. I don't work as a heating/cooling professional, I'm just a homeowner.

Your plumber may be right, and from my experience, it is uncommon that he'd know that (unless, like you said, he was trying to get the home warranty company out of a bill and happened to stumble on the cause).

When your attic gets hot, and there is bad or no circulation, according to Charles' law, the pressure in your attic increases. If it gets hot enough, it can increase high enough that it can suffocate the pilot flame. So if you have no circulation in a hot attic, you can definitely snuff out the pilot flame. Replacing the water heater may not fix the problem. But probably neither would circulating the high pressure air in place with a box fan.

Check your venting situation, especially the passive elements. For example, if you have soffits, make sure they have not become clogged with blown insulation. I would take a blower of some sort (a shopvac that can be reversed for example) and blow them from the outside. Also, look from inside the attic and check what the situation is. If you've had recently blown insulation, it could definitely have caused a problem.

So my suggestion - clean your soffits and evaluate your passive ventilation. Resist the impulse to add active ventilation unless you know what you are doing (they can cause issues). If you can't figure out the problem, it may need the experience of an experienced roofing contractor who understands how to calculate your roofing needs and install additional ventilation. It could be that your two turbines are sufficient, but you are not getting enough air in through soffits and other inlets, for them to pull out. You can google for various formulas to consider how many feet of soffits or ridge vents/passive ventilation you need. If you have no soffits and only a non-functional active fan to push air in, that's probably your problem. I'd add soffits.

By the way, it is possible that as your water heater has aged, it is more sensitive to the increased pressure. This I don't know, I am just theorizing, and someone with practical experience should chime in. Replacing parts or replacing the heater may help or fix the problem, if you are just on the edge of the issue. If you've already fixed the issue, it would be interesting to learn what solution you finally arrived to.