AD&D (1E/2E) can easily cover the 4 needed niches with demi-humans. Truth is, most of the time, the 4 are needed by conventions of adventure design, but those can be avoided by a cautious GM. Experienced players will notice.
BECMI/Cyclopedia plus the gaz rules for General Skills and dwarf priests can also mean a 2 PC party has all 4 roles covered. Elf MU+F, Dw Pr is FI+Cl, and general skills can let them get thief abilities... up to 4 each... for a very constrained story set. But that works well in many published adventures, so I' figured it worth mentioning. Plus, as with AD&D, GMs can simply avoid challenges for missing roles.
D&D 5E has robust non-magical healing options, so it's easily handled.
Any D6 system game. They're trope templates, not roles rules, and characters are broadly competent at easier tasks...
Talisman Adventures. While it appears to have the niches of D&D, it doesn't actually have them mechanically. Spellcasters aren't glass cannons, and due to NPC activation mechanics, it's not a huge issue to be outnumbered if one's got good defensive skills (melee, athletics). And even spellcasters aren't pushovers in melee.