So when you say you want "initiative" to matter more, do you mean the initiative modifier or the initiative process? It sounds like you are talking about the modifier mattering more. The solution to that is easy: more initiative checks, other than just the one to start combat.
My suggestion, for both problems, would be to ditch the Ready action, which I find counter-intuitive, and instead adopt the "On Hold" mechanic from Savage Worlds.
1. Roll initiative as normal at the start of combat.
2. On your turn, instead of moving and taking your action(s), you can go "on hold." Things that go on the start or end of your turn happen now, except for negative conditions that you are suffering that stop or have a chance to stop at the end of your turn.
3. Later, you can take your move and actions whenever you want. BUT if you are trying to interrupt someone else, or two characters on hold try to act at the same time, you roll a contest of initiative to determine who goes first. The winner goes first, and the loser MUST take their move and actions afterwards or forfeit them; you can't lose and then choose to stay on hold. Once you take your move and actions, resolve any end-of-turn things that remove negative conditions from you.
4. If the initiative count comes back around to you and you're still on hold, your move actions are forfeit, and you take your turn as usual (and you might just choose to go on hold again). Basically you wasted a round. Any end-of-turn things that remove negative conditions from you happen just before the start of your next turn.
Advantages of this system:
1. Characters with high initiative rolls at the start of combat have more flexibility in when they act.
2. Characters with high initiative modifiers are more likely to interrupt other characters or preempt them.
3. Players who are indecisive on their turn can just go on hold instead of acting (they neither waste their turn "dodging" nor get saddled with a sub-par "readied action").
4. The Ready action has so many stipulations, caveats, and exceptions that I find it very counter-intuitive, and I've seen a lot of new players have trouble with it, too. This can lead to the feeling that a higher initiative roll is actually bad if you're not in a good position to act when your turn comes around, because Ready is so cumbersome. I don't like the idea that a higher modifier could inadvertently make things worse for you.
This system may increase complexity during game-play if lots of characters use it all the time. However, my experience with Savage Worlds is that players don't choose to go on hold all that often. Usually, taking your turn now is worth more than the chance to interrupt someone later (depending on how high your initiative modifier is). DISCLAIMER: I have not yet tried this approach with 5e; there may be something else about the 5e rules that make this system more complicated than it is in Savage Worlds.