First off, I'll just say that I'm 100% in agreement with your assessment of the problem, and it's not just limited to 5e: 1e-2e were fast even though there were training rules etc., and 3e-4e were super-fast much like 5e.
That said, I'm not quite 100% on board with your solutions, though you've hit some good ideas. Notes and thoughts below...
1) PCs do not advance in level during an adventure. The level you start the adventure in is the level you are for that adventure. (The exception is Levels 1-3. Arguably, 5E characters aren't fully themselves at Level 1, so you can advance up to Level 3 within an adventure).
From this and other parts of your post I gather you don't track xp but instead just have everyone level at the same time. I do track individual xp, and not everybody gets the same every time depending on what the PCs have actually participated in.
However, that's my thing, not yours. With yours, my worry would be that if you're using prepublished modules that expect the PCs to be level x at the start and level x + 2 or x + 3 at the end you might find the start bits of the module are too easy, or the end bits are too tough, or even both; and thus you'll have to do some tweaking.
2) Extended periods of downtime between adventures/levels. The default assumption is that you have a role in the campaign world (usually related to your Background), and that's what you spend most of the time doing. There aren't professional adventurers who are adventuring all the time. Adventures are something you go on when the opportunity arises. Like Bilbo's journey to the Lonely Mountain, or Flint Fireforge letting his smithy go cold.
Excellent - provided you can convince the players/PCs to go along with it; it would be a bit heavy-handed to outright force this on them. Some players see their PCs either as professional adventurers for life, or want to use a quick few years of adventuring to secure their fame and fortune for life so they can go on to other long-term non-adventuring pursuits.
3) Level up two levels between adventures. Adventures start at Levels 1, 5, 7, 9, etc.
See above. Also, I'm not married to the idea of every PC always having to be the same level as every other PC.
4) Give players the flexibility to go on side-quests during the off season, or craft items, or start building a castle or temple or something. Stuff that takes weeks, months, or years. You can get started with that at the Level 3-5 downtime. No need to wait for when the PC is retired and no one cares anymore.
Yes yes yes!
As DM, you can facilitate this by making yourself available on non-session nights for pub or coffeehouse sessions dealing just with one player's character(s). Nobody other than Joe really wants to sit through a game session that's all just you and Joe designing and building Tobias' (Joe's PC) castle, and it's hard to multi-task these things (says me, who's tried it once too often). But if you and Joe go for a beer one night and you each bring your game stuff along, Joe's castle gets built and nobody gets bored.
5) Introduce the trope that there's a mechanism why which the gang gets back together. "Evil is afoot. Old allies need your help. Meet me in Palanthus on the 1st of April."
Another option, that I've done in the past with reasonable success, is that instead of the PCs having scattered home bases try to ensure they end up with a common home base - at the very least all in the same town - and have this become a Known Thing around the region, so as and when needed adventures can in effect come to them.
This has a number of benefits.
A) Fictionally, this seasonable timing is just more believable in a lot of ways. It's actually a suspension of disbelief that someone can become a master of their craft in a month.
Absolutely.
B) It gives the world time to grow and react to the things the PCs do. When they're level 7 they can return to the castle they cleared of monsters at levels 3 or 5 and see that people have moved in and resettled the area. The songs of their heroic deeds have time to be composed and precede them. Their enemies have time to lay and grow more complex plots to stop them. Etc.
Except the PCs aren't dumb; and they'll logically want to proactively press whatever advantage they've gained over their enemies before said enemies have time to regroup and lay new plots. This is where it gets tricky...but there's an answer here too:
Instead of single adventures, make the long break points come between adventure paths such that when they take a break they've finished off whatever main enemy was the focus of the last series of adventures. That way they can take their downtime without having to look over their shoulders all the time, and when the next adventures come calling it's a fresh start on a new story.
Within an adventure path, you can still have (and enforce!) shorter breaks simply by requiring training to advance in level. Training - and the travel to and from such - can easily eat up a month or two per level at low-mid levels; though unfortunately once they get teleport and other long-range transport this advantage goes away. This would also blow up your idea of jumping multiple levels at a time, a trade-off that IMO is probably worth it.
C) No tracking XP! Just don't even worry about it.
I see this as a bug, not a feature.
D) Opens up new parts of the game, like strongholds or extended research and crafting.
Absolutely! (though have a care for your item crafting rules, lest your PCs overpower themselves with the items they make)
E) Allows players to switch out PCs within the fiction. Tired of that human fighter? I guess she joined the Blue Dragonarmy during the years of separation. But look, a cleric has walked into the Inn and needs our help.
This has never really been much of a problem for us. PC turnover is a known fact of life, and players each usually have several to choose from - each of different levels and capabilities, usually.
F) Episodic campaigns. This is preference, but if you share this preference for smaller, more episodic adventures rather than grand campaign arcs, this arrangement is perfect.
You can do this anyway. The main difference is you're putting more time between episodes, which is great.
Reading between the lines, I'm also guessing you want the campaign to last longer overall. Answer here, of course, is to slow down the advance rate. Instead of level 5 just being one adventure, make it a start-to-end adventure path (say, four modules culminating in a BBEG fight which ties off that arc) and have 'em bump to 6th partway through.
Lanefan