Yeah, juggling those early levels (especially when adding stuff) is tough at times. I'll look at it some more and see where they fall. Of course, I might tweak them a bit in the process (big surprise there!).
I've toyed with the 3-class concept: Battler, Caster, Skilled. Is that your breakdown? If you have a doc prepared, I'd love to review it!
Yea, in general (although they're called Warrior, Mage, Rogue). It's a conversion I'm doing of the OSR game Beyond the Wall (which is one of my favorite games ever), just using the more robust action economy and spell lists of 5e.
Basically, you generate your stats and background features using the Beyond the Wall playbooks. The playbook tells you what class you are (Warrior, Mage, Rogue, or a hybrid between 2). Warrior get Knacks and Weapon Specialization, which I've converted into special feats. Rogues normally get bonus skills, which I've converted into, well, bonus skills and Expertise.
Mages are a little different. The playbooks give them some cantrips, spells, and rituals, which I've converted into 5e equivalents. My own change is that magic is intrinsically an external process, you can't simply cast a spell, you have to imbue a focus item and cast through that. So cantrips are cast through a special cantrip wand, which has 7 charges and recharges once per day (either dawn, dusk, noon, or midnight, chosen when the wand is made.) Spells are imbued into special items during a long rest, and recharge once per day (again, dawn, dusk, noon, midnight, chosen per spell during the long rest.) Rituals take no character resources, but are cast from spellbooks and require hours of time and special reagents.
Each class has 6 extra options available at 1st level, from which they pick 2. Each one is keyed to a different ability score, so that a high Intelligence, low Strength warrior is viable, and so is a high Strength, low Intelligence mage.
The main house rules I'm using are: 1) Short rests are 6 hours, and long rests are 72 hours and require a rest at a civilized location. 2) You can use your proficiency modifier in place of your stat modifier for weapon attacks, spell attacks, and spell DCs.