It's rather ridiculous, and is based on the old concept of the "quarterly bill", hence the 3 month limit. The problem is that now it's more the "annual statement" that people tend to get through the post, as modern billing tends to be by monthly direct debit and through an online account service.
Equally ridiculous of course is the solution. Download the online bill that they won't accept, and print it at home on a sheet of A4, which magically makes it sufficiently reliable that it's now accepted. Since modern bills are now produced by printer onto plain A4 printer paper, rather than the "special" paper that used to be used as a means to tell apart the fakes from the original, there is no way now to tell the difference between a scan of a bill printed and posted by the utility, and one you have downloaded and printed yourself via the online account service.
It IS the PDF version you need, not a screenshot of the bill from your online account. The PDF version is what would have been printed and posted out as the "paper bill".
Only a few bills don't have the "online only" version, and these are things like council tax, tax demands from HMRC, etc. However, even this may soon change, HMRC are going to be changing their systems, and the introduction of Universal Credit as an "online only" service will replace many of the benefits that currently come with plenty of "proofs of address" for casinos.
Casinos will be FORCED to modernise their KYC procedures as they will run out of customers able to handle "old school" paper bill based KYC. This moment has yet to come though because many businesses and government departments are still operating their "legacy" systems since not everyone is yet happy or able to operate online only. It's mostly the elderly, but as they die off and are replaced with a new set of internet savvy elderly, we will eventually see online only being the default, and the minority die-hards who still want bits of paper will be told to print it off themselves as it's no longer going to be an option offered by the providers.
Some casinos WILL now accept the original PDF download as an attachment, rather than expecting you to print it off and scan it back to a file. Maybe they know it makes no difference because they can't tell from a mere JPEG image of the document.
However your online account provides the billing, what you are looking for, and what casinos are REALLY after, is the page showing your postal address that would normally be showing in the envelope window of the posted copy, and a clear date of the account statement. They don't care what is being provided, and how much and how you are paying for it. The main problem with screenshots is that they don't show the postal address along with the statement date.
In all of this, the level of security features on the average paper bill has been reduced so that it can be produced as cheaply as possible using bog standard printer paper on a bog standard industrial printer. I was taken aback when I was sent a motor insurance certificate that was devoid of the expected watermarked special paper and fancy security patterning that I had been used to seeing, and finding that it was something I could so easily have created myself and printed onto plain paper at home, and thus get my car "insured" for free. It was then revealed that this move was safe because motor insurance authenticity no longer relies on telling apart a genuine paper certificate from a fake, it's all down to an electronic data check. The same is true of the other paper statements, they can be easily faked, but the electronic checks will tell the fake home creation from the genuine copy of an actual account statement, even though both look the same.
If casinos are still relying on the paper document alone, they are opening up to easier frauds as detection has been moved away from document security features to electronic verification of the data itself.
Even PDF isn't as "locked down" as it once was, and there are free and easily obtainable tools to edit PDF documents, without having to pay hundreds of pounds to Adobe for the official full version of Acrobat that is used to create PDFs in the first place. There may, however, be some security markers that can tell whether or not a PDF sent for verification has been tampered with by the player after having been downloaded from an online account service. This isn't something the player is going to be told, and Adobe may be ensuring that it's something that can be used to uniquely identify and authenticate the creator of a PDF.