I've purchased several houses both with and without mortgages. If you are getting a mortgage, the underwriter will need an appraisal in order to calculate your loan-to-value ratio. They will need title insurance (for which you probably want to get Owners coverage anyway) which requires a title search.
I've never had an underwriter ask for a home inspection report.
But in any case that doesn't mean you need to write a contingency into an offer that the house will appraise for a certain amount or else the deal is off.
Even in "cash" offers there is the offer letter with a check with a deposit amount, but there's still a time period for you execute a purchase and sale agreement and make the final payment. This is where you do things like title search, prepare the P&S agreement, prepare other documents like fire certificate, septic certificate, etc. This always takes at least 30 days anyway, so if there's an appraiser ready to go on Day #2 there's time to get your loan closed before closing day even on a "cash" offer with a quick close.
I'm not quite sure what you mean "no escrow period" but in no case does anyone simply show up at Open House day with a check for the full offer amount and expect to close on the house that day... that's simply not what "cash offer" means.
A "cash" offer with no contingencies simply means your offer amount will not change even if the house appraises for less (requiring a larger down payment to achieve the same LTV) or if there are issues discovered by a home inspector (e.g. roof needs to be replaced).
Note even in a "cash" offer there's still plenty of room to negotiate if non-obvious things are discovered about the house because the seller is still providing a property condition statement (at least in any state I've done business in) where they attest to the overall condition and any adverse issues like septic failing certification or prior insurance claim for flooding or things like that which you discover after making a "cash" offer -- all of it the seller had a duty to disclose and frees you from the offer if you decide to walk away before signing the Purchase and Sale and accepting the quitclaim deed.