I used to believe as you and have been proven wrong in actual play. I still believe it's the case in 95%+ adventuring parties, those without a well build healer and a player who is interested in being that support role - it can be rather unrewarding in terms of spotlight if your fellow players aren't appreciative. But this post is specifically about exploring that corner case, where you can keep up with heavy damage and making sure multiple people don't go down in one round because that would lead to a lot of actions lost.
BTW, I'm also good for avoiding damage in the first place. Assume that will be included in the "support" part and outside the scope of this post.
I have played two separate campaigns without a dedicated support healer, relying on secondary healing (usually paladins or bards), and other support effects. I have also played one campaign in which we had a dedicated life cleric. The first two were much more offensive (replacing the support role with another DPS/tank role), however I should add that many of our characters were also quite self sufficient... So in the right party, you don't need a support character.
For example, any mix of the following characters could get by fine without a dedicated healer.
Moon Druid with/without levels in Bear Barbarian (with is desirable, max 5 levels Barbarian); ideally takes
Sentinel
High AC Wizard with/without lucky feat (with makes for an excellent front-line tank); ideally uses spells like counter spell, wall of force, & hypnotic pattern to shut down enemies
Bear Totem Barbarian
GWM Fighter, Barbarian, or Paladin w/wo polearm master
Grapple Lore Bard
Character with Sentinel
TWF or Archer Assassin, ideally if someone has
commander's strike or another similar ability
Sharpshooter Battlemaster w/wo Crossbow Expert
Paladin/Sorcerer who exploits their increased spell slots with smite.
Fighter/Abjurer or Eldritch Knight - haste/buff builds.
I am sure there are other examples, but you get the idea.
It really depends on what your party makeup is, and how good your party's tactics are. Party's which use sophisticated tactics and which build off each others strategies can usually shutdown and defeat an enemy force without need of a support character.
In one of my campaigns, my 12th-13th level party went up against 10 Fire Giants, 12 Ogres, and 8 Hell Hounds in a single encounter (we accidentally alerted an entire garrison to our presence; this encounter was 2.7x the lethal encounter budget for a party of 7 L20 characters). Our party included my Cleric 1/Wizard 12 w Lucky and AC 23; a BM S&B Fighter with Sentinel and a Shield Guardian; a GWM Frenzy Barbarian; and 3 other equally self-sufficient characters I forget. We did not have a cleric or support character in the party, and despite this - clever use of terrain (the garrison itself) and spells (wall of force) allowed us to tank this encounter, killing everything but 4 giants before we were forced to flee. Had we had our Life Cleric (who was absent that day), we would have defeated this encounter soundly.
I have another character who is an absolute beast on the field (literally and figuratively) - a Moon Druid 3 / Bear Barbarian 5 with Sentinel, and
Insignia of Claws (+1 natural weapons). This character had just acquired a Shield Guardian at the end of the last session I played the character, but for design purposes - this character was designed to be a party shield, but ended up being a high DPS character as well (making them a DPS Tank).
For example - while in Dire Wolf Form and raging, I have 37 HP (resist all except Psychic), Bite +6, 2d6+7? and trip; Pack Tactics; Extra Attack; AC 14; and Sentinel. This character teams up with a High AC S&B Battlemaster Fighter with Riposte (but not sentinel) to make a massive 25 foot front line. Standing behind the BM Fighter, was a PM Sentinel Human Bladelock, to stop anyone trying to get passed the Riposte Fighter. Yes, this group is vulnerable to AoE, but that is what a Counter Spell wizard is for

Sadly, the BM Fighter wasn't a Gnome (as that would have made them even better, with resistance to mental spells).