I agree with both Matt & West, just like in restoration or numerous other professions, there probably are more crooks, hacks and scammers in appraisal/inspection service or antique/classic/used car sales businesses than knowledgeable and trustworthy providers.
Using due diligence to choose one is as important as the subject vehicle itself.
I've probably mentioned it before, but having had brokered or consulted on hundreds & subsequently inspected thousands of vintage vehicles on behalf buyers, mostly because an inability to do it themselves due to living hundreds or thousands of miles away, including on other continents.
Almost every individual has contacted me through word-of-mouth recommendation(s) from those I've assisted before and in 99+% cases have been pleased with it.
As for the OPs dilemma, I can understand why the seller is hesitant and objects to someone bringing their own & unknown lifting equipment.
I probably wouldn't agree to it without pre-approving said equipment, the person operating it and clear terms of liabilities in case of any potential damages caused by lifting.
As far as I'm concerned, PPI* in general is or should be the last thing to do after everything else is done and agreed on, including price**, only to verify sellers claims/disclosures and best if contracted with someone, preferably a shop with necessary equipment, experience & familiarity with the make/model or at least the type of vehicle in question to do it professionally.
And generally the entity performing the PPI, be it an individual or a shop, should be agreed upon by both parties, but usually all costs, including transportation to/from it and other liabilities fall on buyer
* Like with so many other misconceptions in this hobby & industry, there's a difference between appraisal***, general & pre-purchase inspections, but most people don't seem to know or understand what they are.
** Obviously, the price can always be re-negotiated IF PPI reveals notable undisclosed concerns/issues/problems, but trying to obtain & use PPI results beforehand is just moronic in my opinion and likely a waste of time for both the buyer & the seller.
*** For example, years ago, I had to deal with a substantial insurance claim on behalf of a clients vintage car due to a major collision damage and not only the 3-4 of the insurance company's own adjusters/appraisers, but the internationally highly regarded independent appraiser they (the insurance company) also contracted to assess my estimates appeared almost completely inept to provide any sensible assessments of their own.