Cartridges loaded to "Prove" Fiurearms are Specially loaded and calibrated using Pressure testing Devices ( Once Copper Crusher, now Piezo-electric pressure transducers --Direct or Strain-gauge), The cartridge load is now made with a different Powder (usually faster) than the Standard Load; also, Bullets can be of a heavier mass than the standard load, but not always.
ON Black Powder days it was simply a matter of increasing the BP load, and increasing the Bullet weight , where one was at the limit of BP in a cartridge.
IN Smokeless Powder loads, a faster Powder usually got more into the case, as efficient standard loads usually filled a rifle case anyway.
And as mentioned, NO way one can "Home Make" their own proof loads unless they have a well equipped Ballistic Laboratory ( UNiversal receiver, Tunnel, selection of test Barrels, etc, to work up the Proof load in the cartridges first, to eventually use in the subject Firearm.
"Proofing" has been with us since the Middle ages, and Formalized by the British Gun Proof Act of 1813, and by similar legislation in Belgium in the 1830s, and in France Earlier. Germany regularised its Proof Laws in 1891, although it had proofing regulations earlier than that.
THis was basically a form of Commercial protection and consumer protection. AS well, the Military of all nations demanded "proof of suitability" of all items before acceptance into Service. Military proof and Civilian (Commercial Trade )Proofing were similar in concept, but differently applied.
With the development of steels etc in the 1890s forward, The Individual proofing done in Europe was for several reasons (a) It had always been done before;(b) it really did check the structural integrity of a gun mechanism (c) in breechloading firearms, it "seated" locking surfaces in case of any minor manufacturing irregularities.
The US, with the Springfield Rifle M1903, introduced "Statistical proofing" where one rifle every 100 or so was selected at random, and subjected to various acceptance tests ( ie, accuracy, 30% OPP ( over-Pressure Proof, or "Blue Pill" cartridge) and so on. ( Read Brophy, "The US Springfield M1903 Rifle" for details). European countries continued individual acceptance proofing of all rifles ( the Germans especially).
These days, with higher standards of QA in Gun manufacture, steel composition etc, one could opine that "proof" is immaterial, at least for New Made guns...a view not shared by European nations...which have unified their proof regulations (Britain is still independant, and does not reciprocally recognise CIP ( Eu. Community) proof marking, requiring re-proof at Birmingahm Proof House before Commercial sale in Britiain.)
Re-proofing of Used firearms is required upon entry into a country in Europe and the UK...even if initially proofed in another Eu Country.( there are some exemptions under the Schengen Open Border accords in Europe).
SO, getting back to the original question...why proof? ( or make your own proof cartridges?): Personally, I think that a string of 5 or ten shots of Standard ammo should be sufficient to tell if a firearm is OK or Not, by checking the resultant shell cases etc.
( of course, done remotely, as described by many posts on many Boards.).
IN Europe, it is a simple matter to apply to the Proof House to "re-prove" your firearm (it is obligatory for all New Fireams, and in some countries, for any major repairs, such as Barrel replacement or calibre change). The trouble is ( as happend to shotgun owners in Britain,) if you send one of your Fine Purdeys to be reproofed and they find the barrels fail "Technical" proof ( "out of Proof", for wear etc) they chop the barrels before returning the gun to you....Ouch!!! if the gun, even as a collectible wallhanger, is worth several Thousands of Pounds....
SO, best stay way for an area which to the untutored, is fraught with dangers (To yourself and others). If you really need to proof test a gun ( whether new or old) have it done by a professional, although in the US these are few and far between, as the US doesn't have any National proofing Law or system...it relies on Litigious lawyers to supply the necessary "consumer protection" ( after the fact)...some Gunmakers still proof, but the majority used the Statisatical sample method, as equally valid.
Regards
Doc AV