It is unusual, but not unheard of, to do a quiet title action in connection with an arms length, non-related party sale of real estate.
The far more customary practice is for the seller to do the quiet title action (at the seller's expense) if it is necessary for the seller to have what is called "marketable title" to the property, before it is even listed for sale.
It would also be more customary for the closing to be extended to a date after which the quiet title action can be completed, and for you to lease rather than buy the property prior to closing.
This way, if the quiet title action fails to quiet title to the seller, the closing with you doesn't have to be undone, and you can just move your stuff out with reasonable notice from the landlord-seller, and you can look for somewhere else to live or can try to cut a deal with whoever is determined to be the true owner of the property.
Also, as a practical reality, it would be quite unusual for a mortgage lender to agree to provide you with a mortgage without title insurance in place. I very much doubt that the deal can go forward as planned, even if the you and the seller agree, if you need mortgage financing to buy the house.
This said, there is nothing particularly unusual about a title insurance company's requirement that a quiet title action be completed in a case where the current owner took title via a tax sale from a deceased owner's probate estate. Any irregularity in the tax sale process could vest the property back into the estate of the deceased owner, the tax sale buyer likely paid less than fair market value, and there are probably special notice requirements involved in a tax sale from a deceased owner's estate that don't apply (and extend the statute of limitations for contests of the tax sale) that wouldn't apply in the case of a tax sale from someone who is alive. Some of the potential irregularities wouldn't appear in the public record or in any other documentation that you could demand (e.g. a forged signature or an error in crediting payments of taxes to the wrong account).
The likelihood that a quiet title action will vest title in the name of the seller is high, but the fact that the title company is not willing to insure title in its current state is strong circumstantial evidence that the risk that the seller does not have good title to the property is real.
Your concerns are not unreasonable, and the safer course of action would be to restructure the deal so that you do not take title unless and until the title is quieted in the seller in a lawsuit conducted at his sole expense and risk, even if you move in pursuant to a lease pending that process. The existence of an (amended) real estate contract between you and the seller should be sufficient to protect your interest in the deal that you have struck between now and closing. It is possible that your financing could fall apart between now and closing, but often the circumstances that would make that happen are circumstances that would cause you to wish you never did the deal in the first place anyway.
When I represented some heirs of a decedent in a similar case (involving a fraudulent sale to avoid a tax sale, rather than a tax sale itself, from a probate estate), the deal ultimately struck was to have the heirs sell the house to the buyer who was under contract, with proceeds split between the nominal owner of record and the heirs who might have had a claim to undo a prior sale. Such a resolution if there is a contested quiet title litigation, in lieu of taking the quiet title dispute to trial, is another resolution that would be fine with you if the closing is postponed until after the title dispute is resolved.