Why would you want to make ranged attacks even more powerful in 5e?
Calculate the average turn order a ranged attack would have gone at under the old system, which included the full dex bonus. Now compare it to this new system, which does not include their dex bonus. I think you will find they're going roughly the same point in combat that they always were, relative to the str-melee combatant. I don't think they're really made more powerful - they're just about the same amount of powerful from this system I suspect.
So just as a guesstimate, a two-handed fighter would typically wear heavy armor and dump Dex, as it's only purpose would be for Initiative, and dex saves. Initiative isn't enough, on it's own, to incentivize that player to increase their dex over Con or the all important Wisdom save, and Dex saves tend to be the least impactful for that fighter (because they impact hit points, which they tend to have more of anyway). So, let's assume the Fighter has an 8 or 10 in Dex (-1 or 0), which I think is fair.
The Ranged Attacker, however, will max dex. Their AC depends on it, their attack bonus does, and their damage does. So they will likely have a 16 to 18 dex (+3 or +4), and eventually a 20 (+5).
Old system: Ranged attacker rolls a d20+3 or 4, so an average of 13.5 or 14.5. Melee attacker rolls a d20-1 or +0, so an average roll of 9.5 or 10.5. It's roughly a 4 point bonus the ranged attacker gets, and eventually a 5 point advantage once the ranged attacker gets a 20 in dex.
New system: Ranged attack rolls a d4, so average roll of 2.5. Melee attacker rolls a d8, so average of 4.5. So the Ranged attacker has a +2 bonus on the Melee attacker.
I am not sure of the percentages on that second one, but to me it seems roughly pretty darn similar. Seems like the Ranged attacker has roughly a 20-25% advantage on initiatives in both systems.