Thanks for the additional info!
When I say I want to keep counterspell I have been misunderstood. I want to keep the ability to prevent a spell. I do not want it to be a reaction. I want it to be an action. As for the other abilities like uncanny dodge I would be looking for suitable replacements sans interruption.
Tracking initiative is a problem. I have tried many different methods. Some work better than others. I still feel that this mechanic exacerbates the problem without actually adding something of value. Instead it adds dice rolls and extra decision points. These tend to most penalize players without system mastery by increasing delay between turns.
Some things are non issues. Like legendary resistance (or even shield) It doesn't require a new roll or a target at best it just changes the outcome of one dice-roll that has already been made.
Let's address Counterspell first then. How would you resolve this? As an action, a player states "I am countering their next spell."? Do they know what the spell is? Are they allowed to do it as an action on their next turn or must it be preemptive? Would the target of the counterspell know it was cast against them? How you answer those questions determines how you want it to work.
For Initiative, are you rolling once for the entire encounter or do you roll each round? If you are rolling each round, thinking the variability it offers is good, I can tell you from my own experience it isn't needed. We changed to rolling once a while ago and have had no issues with it.
Our DM runs our games from his laptop, linked via HDMI to a 50" wall-mounted TV which he displays maps on and stuff. One thing he displays is the Initiative order so anyone can look up and see when their turn is coming.
A less-tech option would be once the order is determined, hand out cards to each player with the name of the person before they go. This way, when that player or DM is acting, they know they are up next. We've done this before and it worked well for us. After the encounter, the cards are returned to the DM until the next one.
When a reaction is used, that character/creature acts, and then it resumes to the current person's turn.