Using 1e with maximum-hp-at-1st level, and at least half max at higher levels, solves that problem.
4e PCs can die in one hit. IMC a 1st level Ranger was brought to 1 hp off negative-bloodied (insta kill) by a dire wolf's attack, in the first round of the first combat of his first session. Plenty of low level 4e creatures have critical hit damage enough to insta-kill 1st level PCs.
The big difference from 1e is attrition; 4e PCs are expected to start each fight at full hp and to lose lots of hp during the fight. IME 1e PCs will often begin a battle at below full hp.
This means that in 4e weaker encounters are much less threatening to the PCs, because they do not much heighten the likelihood of the group dying later. Conversely stronger encounters become more threatening to the PCs because the chance of a TPK goes up much faster with increased enemy strength.
In 1e, lots of weak encounters serve to attrite group resources over time, while going by quickly, and providing little XP. In 4e lots of weak encounters makes for very little threat, provides a lot of XP, and takes a long time to play.
Given the choice, 4e PCs have a strong incentive to seek out encounters below party level and avoid encounters above party level - this maximises XP gain while minimising threat. 1e PCs have incentives to seek out stronger encounters as only stronger encounters give significant XP, and they often mean more treasure.