Yes, readying a spell behind full cover would prevent counterspell
Counterspell depends on sight and a clear path to the target
Counterspell has a casting time of:
1 reaction, which you take when you see a creature within 60 feet of
you casting a spell
That means that an opposing spellcaster must be able to see the intended counterspell target. Additionally, counterspell must follow the general rules that spells must also have a direct path to the target. Full cover should prevent either or both of these requirements from being met (depending on the type of cover used).
Readying a spell behind full cover and releasing it will prevent counterspell
Casting a spell while under full cover follows all of the appropriate rules.
You can cast a spell behind full cover even though you don't have line of effect per Jeremy Crawford:
The intent is that your target must be within range when you take the readied action, not when you first ready it.
And your trigger "when I have line of sight to the enemy mage" is perceivable.
Then, when you release the spell, it cannot be Counterspelled. Again clarified by Jeremy Crawford:
Counterspell foils the casting of a spell, not the release of a spell that was cast previously using the Ready action.
So, there is no reason why this would not work.
The trick will be in getting an available and convenient source of full cover.
Opportunity cost1 and downsides
Since you are spending your reaction, that means that this strategy will prevent you from casting counterspell as well as any other reactions until the start of your next turn.
Also, you must concentrate on the readied spell which means that you will have to drop concentration on anything else you might have been concentrating on. Also it opens up the possibility that your concentration is lost due to opportunity attacks, enemy readied actions, etc.
Is it overpowered? No (Fool me once...)
At first glance this strategy might appear to be overpowered. However, like most strategies, it has counters. You might get away with using this once, but a smart enemy (especially one that has access to counterspell) will not fall for it for more than that. They can move to negate your cover, move behind cover themselves, prevent you from moving behind cover, and/or use readied action to enact any of the above and more. In the end, this strategy should only be overpowered if creatures aren't responding intelligently to it.
There are other ways to prevent counterspell too
It is also worth noting that making yourself invisible (which is not incredibly difficult at higher levels) or blinding the opposing caster would also prevent any counterspelling without many of the disadvantages of the readied action technique.
Additionally, if the terrain and spell allows, you can cast outside the 60' range of counterspell.
1 - Thanks @Slagmouth!
Best Answer
The Ready action is described on PHB, p. 193 or here in the Basic Rules.
You may set a perceivable trigger. The rules are somewhat vague in two aspects:
One might assume, that this means a single trigger, not a bunch of triggers.
It also has to be perceivable, so if you wait for someone to come out of cover somewhere in front of you, you cannot possibly perceive an enemy behind you by common sense. Although the rules do not explicitly say so.
If I had to interpret the rules strictly by the rules and English grammar, there would only be one trigger allowed. The trigger. You would have to pick one of your two.
The way my group plays, we allow multiple triggers if all are perceivable with the same senses and in the same direction. So if you wait for someone to step out of cover and someone in your line of sight attacks you, your ready-action would be valid. But you should consult with your group if they think it's a good way to read this rule.