Is my routine OK?

It might be useful for you to think in term of planes of movements and then look for a push and pull exercise for those planes. (You have most of them covered....)

Vertical upper body - shoulder press, lat pull down (there's your push and pull)
Horizontal upper body - chest press, seated row (ditto)
Lower body - leg press is your push but a pull would be useful (is there a leg curl machine perhaps as you want to stick to machines?)

Many beginner routines start with full body 3 times a week majoring in the compound lifts.
Which is what you have done. :flowerforyou:

"Virtually impossible" last rep is quite probably too high an intensity for you. Hard would be enough and generate less fatigue, going to failure too often isn't a great idea and isn't required to make good progress - especially for a beginner. Two in the tank is one way of thinking of it. Less fatigue means more complete recovery and lower chance of impacting your next workout.

You're completely missing the posterior of your lower body. You need something like deadlifts or some version of deadlifts in there...I'm personally not a big fan of leg curls (or leg extensions) unless you're rehabbing. With what you're doing I'd probably go for Romanian. They can be done with barbells, kettlebells, dumbbells, and my gym has a cable machine that you can use for RDLs as well as deads.

Since you're using machines I would recommend as much as possible to use cable machines as they provide more free range of motion rather than a fixed path. For progression, you may want to consider something like 3 sets of 8-12 where 8 is challenging and when it becomes less so you up the reps until you get to 12 and then increase the weight...in my experience, that has allowed for greater overload progression. It is also beneficial to work in different rep ranges, so something like that would automatically build in that work along the way. You also don't really want to train to failure as in...

so that it is always virtually impossible to do the last rep

It's just not really an optimal way to train. You want it to be challenging to where you should have 1-2 reps left in the tank but not at failure on your last rep.