Your solution is a fair and effective one. It does rub me the wrong way since decoupling resource recovery that way just seems odd.
For example.
If the current state of affairs is: Travel out and adventure, return to town, rest a week and repeat, if long rests are a week long it makes sense logically, the characters spend the week recovering and resting.
If I make it the end of every other session, then I have to figure out why characters can long rest one week and adventure but can't long rest the other. It is counter-intuitive. If they can rest the first week, why can't they rest the second?
No, you just make 'resource recovery'
not linked to in game resting.
You seem to do this anyways. You grant a long rest at the end of your sessions (at which point the characters 'in game' and the players 'in real life, take a week off) automatically.
If that's too jarring for you, simply rule that the 'end of session rest' works as follows:
The first and second in game 'end of session rest' grants the benefits of a Short rest. The third such in game 'end of session rest' grants the benefits of a Long rest, and then the rest counter resets to zero (and the cycle begins again).
So in game the players characters are resting after every game session, binding wounds, taking time off etc, and gaining a benefit (short rest recovery of powers, healing via Hit Dice etc, PCs with the healer feat get that recharged etc). Every third such rest is a Long rest.
Be
very clear to your players before you do this that you're doing it, and why, so they know NOT to nova. Warn them that nova-ing could lead to a TPK later on down the line, and advise them to marshal resources accordingly. If they Nova anyway, screw them - dont listen to their whining for a rest, hit them with both barrels so they know you're not messing around (but avoid a TPK - you want to get them self policing resources around this paradigm, not murdering them).
On your part,
tone down your encounters if you do this. Dial them back to Medium-Hard (with the occasional easy and deadly encounter as well). Your players should get around 2-3 encounters per session (followed by a Short rest at the end of the session) and have to deal with 3 such sessions (of 6-9 encounters) in between long rests every 3rd session.
You'll find class balance will straighten up, and Monks, Fighters, and Warlocks will suddenly hold their own with the Paladins, Barbs and full casters. Every session the Locks, Monks and Battlemasters come to the table with a fresh use of Second wind, Action surge, Pact Magic Spell slots, Ki points, Superiority dice and so forth.
The full casters (and paladins and barbarians) have to marshal their resources over 3 entire sessions.
At 5th level, this means your casters have 4/3/2 slots each for 3 sessions (3 spells per session), your Barbarians have 3 rages (enough for 1/ session) and your paladins have 2 x smites per session (4/2).
That brings those classes roughly on par at this rest frequency with the Short rest heavy classes.