"the disbursements from the charity would also be considered tax free (because charities don't pay taxes)."
That's also not quite right, and not nearly that simple. The tax status of charities and the tax status to a beneficiary are separate issues. If a legitimate charity decides for some reason to pay for a new roof for an entity that's not needy the gift may not count as taxable income to the recipient, but the tax exempt status of a charity is subject to a bunch of rules. One of those rules is that the charity can't unfairly benefit any director, officer, or private individual. If your "charity" makes plenty of legitimate donations to worthy causes the occasional questionable donation might slip through or it might not. Also, if your "charity" is mostly funded by you it will be a private foundation, and that means there's even more scrutiny in regards to self-dealing and personal benefit. If you don't already know that Donald Trump's private foundation has run into a few problems you might look into it for some insight.
"*There's actually a loophole for this. If the foundation paying for repairs to your home rents it for the period that it takes to do the repairs, the homeowner doesn't have to pay taxes on the gift of a new roof. This loophole is used by those TV shows that renovate peoples homes so the homeowner doesn't get a tax bill for the value of the renovation."
The result may have been the same, but that's not quite how it works. If your residence is rented for less than 15 days per year the rental income isn't taxable income. That would mean that if the shows paid rent in the form of improvements to the residence there's no taxable income to the owner, at least in theory. The IRS has specific rules about renting or selling things for less than market value to address disguised gifts, but AFAIK they don't have specific rules about renting or selling something for more than market value.
Despite what the law and rules are I'm a bit surprised they could get away with that, because it certainly skirts the line between tax avoidance and tax evasion, but in the case of the TV shows and homeowners there's (presumably) no prior relationship between the two legal entities involved. If the charity rented your home for legitimate uses at a fair market value you probably wouldn't have any problems, but there's not that much financial benefit in that. OTOH, if your "charity (or private foundation) pays something well in excess of fair market value and the IRS comes looking the personal relationship could bite you in the ass.
Of course you might become an ordained minister and start a church. That scientology can charge their members for things makes me figure that with a bit of creativity and a good legal team the possibilities are unlimited. After all, who wants a bunch of strangers to keep walking through the doors of your beachfront church?