I find it nearly impossible to justify anyone going from 1st level to 20th level in 7 weeks. Anyone in the game world is going to see the PCs do that and want to duplicate it. I mean, how can you begin to argue that a 1st level wizard just out of his apprentice diapers is casting time stop and meteor swarm in 7 weeks? It's simply ludicrous!
I understand the PCs are the heroes of the story and that the rules for them don't apply to NPCs, but still some amount of internal verisimilitude is necessary in order to have immersion and suspend disbelief.
My current 3.5 campaign is in its 7th real life year of running and the PCs are just hitting 18th level (we play every other Saturday). I did this by:
1. Halving XP rates
2. Enforcing time off for the PCs (i.e. you simply cannot adventure and put your life on the line for week after week without taking a break...you will collapse emotionally and physically and mentally)
3. Having the PCs have to take time to do things like sell or buy magic items
4. Greatly increase the magic item crafting times (45 times longer!).
I guess I do #2 and #3 in my 5e game, though PCs can't generally buy or sell magic items IMC, there's not much of a market. Not any routine item crafting either, so no #4. I haven't reduced XP awards but I run an online text-chat game which is a slower format, PCs who have been there from the start are 8th level after 39 sessions. 39 game sessions has been just over 3 months of game time, campaign first session was 20th March 2015 so 6.5 months of real time.
I think your approach is more necessary for 3e/Pathfinder than for 5e. 3e/PF is the only system
where the default rate of advancement really does feel silly-fast to me.