One of the things that made Captain America believable is that he's a "one off". As in, since he was created, they were never able to do it again.
The main scientist who did this dies soon after his creation, and from the plot of the movie, we do know that he'd been specifically looking at Steve to create it--so it's possible that what he created was very specific to Steve, and that there actually isn't a "one-size fits all" solution to the problem of creating a super-solider in this world or in ours.
I do have some problems with the premise, because it doesn't actually fit what Steve is. You say:
a being with super strength, speed, whose muscles never tire due to lack of lactic acid building up in them, and all other human abilities enhanced with none of the drawbacks?
But there is evidence in the movies and some of the comics that this is not true, that this doesn't track with how Cap actually works--and rightly so. You can't build muscle without destroying it a little. It's those tiny tears that make you stronger. And Steve does tire, it just takes more than it does for a normal human. He's also not super fast.
Getting tired is a function of any biological body, it's just a matter of level of stamina. It's not possible to create someone who never gets tired. Steve sleeps, maybe not as much as a normal person, but he does. And he needs to eat. Without these "draw backs" you would have an entirely static individual who would not be able to improve their strength or stamina.
But if I were making a super solider, there's one single thing I would do--if it were possible (which it is not at this time without nasty-side effects)...I would make them better at using and gathering oxygen.
Or I would specifically train them using an oxygen boost
Paratroopers had to be in excellent shape then (and now) to withstand the rigors of falling out of the sky to who knows what but this mission was so important the general called for an extra 20 days of super intense training.
Bradley tells us that they used supplemental oxygen to breathe while riding stationary bicycles and got pushed way, way, way beyond where they had ever been pushed before. No pills, nothing but oxygen enhanced exercise and rest. Bradley says “it was incredible. I was already in fantastic combat shape but after this training I was in the best shape ever by a very wide margin." source
Any super-solider program that's realistic will take actual work, time, and it's likely going to be a multi-discipline body science type thing, looking to push regular men a little beyond the norm, but not necessarily to the levels you are looking for. I would also argue that like athletes, they will have a limited window of time in which they could do this--age is a problem.
What is happening in the real world is more..technologically based, which comes with its own issues.
My answer to your base question, as it stands is no. Not without any drawbacks.