I do like it, and think it would be better than the default rules.
My preferred method would be something like how 1e/2e does it, but that is probably not very popular.
We currently do old-school 1E/2E multiclassing and it works great, but it
is very powerful and leads to
very strong combinations! Use with caution!
Some differences to note:
(THIS IS ONLY TO OUR CURRENT HOUSE-RULES, not the idea of this thread.)
You have character level and class level. You
must choose your primary class. This dictates your skills and saving throw proficiencies. You gain one skill for your secondary class and one of their saving throw proficiencies.
Your character level is determine by total XP. Your character level determines your HD, proficiency bonus, and when you gain ASI (4, 8, 12, 16, & 19).
Ex. A character with 36000 XP is 8th character level.
Your class level is determined by dividing your total XP by the number of classes you have.
Ex. The above character with 36000 XP would have 18000 in two classes (making them 6th level each) or 12000 in three classes (making them 5th level in each).
Suppose for this example we made a Fighter/Rogue. He would be Fighter 6/ Rogue 6 (CL 8). The (CL 8) tells the player he is 8th character level. Selecting Rogue as the primary class gives 4 skills from the Rogue list and proficiency in DEX and INT saves. Having Fighter as the secondary gives 1 skill from the Fighter list and your choice of save proficiency in STR or CON.
This character would have all the features (including choosing subclasses) of a Fighter 6
AND a Rogue 6. Now, as features go, that RAW would require a 12th level character! So, you can see how having features akin to a 12th level character but at level 8 is powerful, right? True, you don't have as
powerful individual features maybe, but the collection of more weaker features is very strong.
Now, this character would have proficiency +3 (8th level, remember?) and two ASIs (4th and 8th levels). As a Fighter 6, you would gain an additional ASI as your fighter feature. We play the bonus Fighter ASIs and the bonus Rogue ASI
MUST be used on a feat, however. So, this character would gain a feat for Fighter 6.
In 1E/2E multiclassing, HP would be 6d10 (fighter) + 6d8 (rogue), averaged out. We do it differently (of course, LOL!).
You gain HD according to character level and medium-sized characters gain d8's. So, you would begin with 8 at 1st level, and having 7d8 more by the time you reached character level 8. (Small-sized creatures get d6's, sorry.

). You apply your CON bonus to each HD as normal.
Your HP are modified by your classes. Each level of Barbarian adds +2; and, Fighter, Paladin, and Ranger adds +1 to HP. This sample character is Fighter 6, so gains +6 hp for those fighter levels.
Each level of Sorcerer and Wizard
subtracts 1 hp per level of those classes.
While this probably seems confusing, it works well because we were running into the problem of having strong features (requiring more powerful foes), but not enough HP to face them. This way there is a HP boost that makes them viable to face stronger enemies.
Summary:
Fighter 6/ Rogue 6 (CL 8)
HP: 8 (level 1) + 7d8 + 8 * CON mod + 6 (fighter)
HD: 8d8
Skills: 4 Rogue + 1 Fighter + background as normal
Saves: DEX, INT, (STR or CON)
Proficiency: +3
ASI: 4th and 8th, bonus feat at Fighter 6
Once again, this system works very well for balance, but makes very strong characters and for a power-gamer style table. It is fun, but takes some real balancing when creating encounters.
If you want further clarification or another example, please let me know.