Most 5e house rules I've seen add complexity to the system. However, I'm broadly interested to know what other DMs have done to simplify 5e. FWIW, I'm perfectly comfortable running it as is, just picking brains for material as I work on a side project. Some examples might be an even simpler initiative system, changes to the action economy (handling of action/bonus action/reaction), reduced character creation options, etc.
There is a lot of complexity, but it's hard to jettison without depriving players of options.
Obviously, first, don't use any optional rules. Feats & MCing are just added complexity.
Bonus actions and concentration add complexity, for instance, so do reactions - removing everything that uses them would reduce the complexity of the game. Not removing the mechanics (which might render a variety of things OP or otherwise broken), but everything that uses them - so TWF, healing word? use bonus action: gone. Haste, Summoning? use Concentration: gone. Opportunity Attacks, Delay? use reactions: gone.
Sub-classes add complexity, you could pick the simplest/most-iconic sub-class of each class, and use only it for all instances of that class. Fighters'd all be Champions, Rogues all Thieves, Clerics all Life-Domain (healers), etc.
Multiple classes with similar concepts add complexity: The Barbarian & Ranger are redundant with the Fighter, the Paladin redundant with both the Fighter & Cleric, the Sorcerer, Warlock, & Bard redundant with the wizard. The Bard & Monk redundant with the Rogue. In other words, cut it down to the Big 4 classes. (I'd say cut the Rogue, too, and give all it's out-of-combat toys to the Fighter, but that's not a popular opinion.)
Mixing backgrounds & classes freely adds complexity. Peg each Background to a class, and have members of that class pick only from those backgrounds - much like a 2e Kit.
Having both saves & attacks adds complexity, standardize on one. Better yet, take all rolls to one side of the DM screen - either the player always rolls (when the DM calls for a check), or the DM makes all checks and the players needn't even necessarily know stats.
Six different saves adds complexity: just use highest mod + proficiency for all saves.
Many different weapons add complexity. Give each type of weapon (by Combat Style, essentially) a set of stats, and use them for all such weapons. All two-handed weapons use the "two handed weapon" stats.
All projectile weapons - bow, crossbow, sling, M-16 - use the same stats.
Preparing spells adds complexity - convert all casters to known spells only.
Casting spontaneously still adds complexity (full on analysis-paralysis from the many choice of not only any known spell to cast, but what slot level to cast it at!) - convert all casters to knowing a number of spells equal to their slots, and casting each one only once per day.
Y'know what, that's still too complex, give them, like one spell per Tier.
Y'know what, just cut the casters entirely.
Champions & Thieves will do fine on their own.
There, /that's/ simple!