Well, I have some lag too but I don't think you need to wait for the ninjutsu icon which is not always reliable.
Sometimes the Ninjutsu action button doesn't work if I press it once, like in the OP.

I just do my mudra in order and press the ninjutsu button a few times. Then I queue up the next action.
Ninjutsu still goes off and the next attack goes off soon after while both the ninjutsu icon and character animation catches up.

It still eats into the GCD, but waiting will take even more time.
You can press the ninjutsu action a few times with no consequence. It's not wasted time either since a few presses is faster than finishing the ninjutsu, so my next action is already queued before it's all done.

For keeping everything up, it's not so bad even with lag.
The lag is just annoying because it feels slower than it should.
I refresh Fuuton if my ninjutsu is up and the buff has <20s.

The rest is as Pun and Dark has said.
I'm fairly new too but I've already gotten a feel for when my dots/debuffs need refreshing without really monitoring them actively.
I used to main a Monk whose timers are mostly built into the combo.
NIN is asymmetrical but still has a rhythm (like DRG, I guess).

Latency shouldn't cause downtime unless you mess up a Fuuton cast because of it.
Just do the combo in order and smash the ninjutsu button a little and it shouldn't happen though.