I come from a balanced position on the extreme.
1) I do think 5e's monster have a "handicap" in being terrifying because of the loss of those crippling abilities.
2) Good DM play can overcome those handicaps and still create memorable and fun fights.
A few suggestions on how to do it:
1) Surprise is huge. Surprise is a huge swing factor in a fight. I have said many times I will take a fight with surprise than adding 4 CR to my fight, its that strong. When my party gets hit with surprise, I always see them sliding to the edge of their seats.
2) Catch the party out of the norm: If you have monsters running at the party with the meatshields in front, everyone rested and refreshed....than yeah they are going to get pulped in short order. However, attack them in the tavern when they are in the bath, have someone sneak up on the spellcasters while the fight is ongoing...and the fear factor rises quickly.
3) Make a single party member the target, no matter what. Have a hoard of monsters go right for party member X....taking AOOs and whatever else is needed. This is both scary as that player knows flat out that they will attack him until he is paste, no mercy. But on the flip side it can be a rewarding challenge if the party members find a way to protect that person, or buff him up so that attacks against him are weaker than if they had gone after someone else.
4) Terrain is the key. Terrain is as much a monster as anything else in the fight, maybe more so. An ooze is just a blob of hitpoints. An ooze that attacks the party when they are squeezing through a tight cavernous passage becomes a terrifying threat.
5) Alternate Loss Conditions: Death is not the only consequence. Use timed encounters, the party only has 3 rounds to get to the artifact before the ritual goes off. The party is protecting a 1st level character, so while they can take the pain, their ward cannot, etc.
6) Use an overwhelming threat...in the background. Have a purple worm erupt 200 feet from the party, but not engage them. Have a Nightwalker be seen on the horizon on a dark foggy night. For some people, just the image of what's out there can be scary.
7) Reflavor abilities. If players are used to certain monsters, change up their abilities to make them strange and creepy. My best example. I took hobgoblin stats and gave it Magic Missile at will (just the 1st level ability, nothing too extreme).
I reflavored it to this horrific black monster with a huge mouth and teeth, that was dead silent. All it would do was point its "claws" at the party and swipe....and then immediately the party would take damage (aka magic missile). They freaked out! "What do you mean there was no roll....it just hit me? No save no nothing!!!" And then....I gave them 6 of them to contend with. One use of magic missile isn't so bad, 6 copies of it flying at once is devastating. The party ran scared, and came up with all sorts of ambush tactics to take them out. All from a creature with low hitpoints and the use of a 1st level spell.