A few thoughts on the enclosed chain case:
The protection factor is obvious. If you're riding that bicycle as your daily transportation, 365 days a year, no matter what the weather. you want this. Especially if you're wearing something better than jeans to ride in. You can go virtually forever without re-lubricating your chain, once you've oiled it up in the first place. And you can way over-lube it without making a mess of your bike or clothes.
The style factor is unbelievable. Nothing says "antique bicycle" better than a fully enclosed chain case. In my time in the reenactment hobby, I've had a chance to work with a few World War I, World War II, and Depression-era groups that utilize a bicycle in their group impression. While using an actual antique is obviously the way to go, I've worked with individuals who backdated 70's roadsters and English racers as fill-ins until they could find a proper bike. And, for that matter, used the fill-in for weekend tacticals when the weather or conditions could damage an antique bicycle. In these cases, going to an enclosed chain case is mandatory, just to cover up the fact that the frame angles aren't slack enough, you've got cable operated brakes, etc.
Yes, they are a pain to work with. Now, I've yet to work with an original pre-WWII roadster that came into my hands with the enclosed chain case intact. My personal experience is with the knockoffs. Primarily Chinese Flying Pigeons. And the knockoff Chinese and Indian bikes have chain cases that are, maybe, 1/16" wider than the chain. Which means, since said bikes are invariably going to have a bit of warpage in the chainwheel, you're going to get some degree of scraping in your pedal rotation, no matter how much you adjust. It's maddening, and I can only assume that the original British/Dutch/Belgian/whatever bikes had higher quality chainwheels that weren't warped. On the other hand, once you've removed and reinstalled a chain case 3-4 times (usually trying to adjust the scraping out) you get used to it, and it's no longer an effort. You know the method.
Yes, if you get a puncture, you've got a problem. My understanding for this (and it's just an understanding, the only flat I've ever had when I was a distance from home was the complete failure of my tire's sidewall - call mama to bring the van) is that you don't remove the wheel. You lay the bike down with the wheel still mounted, dismount the tire from the rim (28" rims actually make that kind of easy), use the frame pump to find the puncture, patch it right there, and remount the tire and reinflate. I'll assume it works, one of these days, no doubt, I'll find out since my Raleigh Tourist is one of my two daily commuters.
From my days at the Raleigh shop: The Tourist, in England, came with a full coverage chain case. The American models came with a hockey stick chain guard due to weight. Not weight as in "affecting performance" (nobody will ever use the term performance regarding a rod-braked roadster without chuckling) but something to do with shipping weight, or import duties based on weight, or something else along those lines. My old boss was never clear on the details. Suffice to say, the bike with a hockey stick chain guard barely made it under the weight limit, while the enclosed chain case version was definitely over.