Absurdly old comment, but to answer the question it's because of the Montreaux Convention to allow a ship through the straits. If it's got heavy anti-ship missiles on it, you have a decent claim to being an "aircraft-carrying cruiser" not an "aircraft carrier".
This is a misconception. It has anti-ship missiles because the Soviets didn't view carriers the same way the West did. They were not for strategic nuclear attack on the enemy homeland. That is what ICBMs are for. They were for sea control and area denial by extending the range of surface to surface missiles and the air defense umbrella of land bases, to defend the SSNs patrolling the bastion edges, and literally nothing else.
The Soviets would have been allowed to move the Kuznetsov for the same reason they were allowed to move the Moskvas and the Kievs: they were transiting from their shipyard to the home port and there's no particular reason to stop them from doing so. So Turkey didn't. The only time those articles come into force, as we've seen, is when there is a war involving Black Sea powers. It's why Russia has been having to negotiate with Turkey for access to the Black Sea and transiting of equipment from the Far East and Northern Fleets that is nominally shouldn't be allowed to move. That is how the initial grain deal happened.
In peacetime, the naval forces of all States, whether they have a coast on the Black Sea or not, are under the obligation to give prior notice to Turkiye through the diplomatic channel before transiting,37 except for “naval auxiliary vessels specifically designed for the carriage of fuel, liquid or nonliquid”
on condition that they shall pass through the straits singly.38 While the Black Sea countries should give notice eight days before they pass, this period can be extended up to fifteen days for non-Black Sea countries. Without being under any obligation to stop, the commander of the naval force shall communicate to a signal station at the entrance to the Dardanelles or the Bosphorus the exact composition of the force under his command.39 The maximum aggregate tonnage of all foreign naval forces that are in the course of transit through the straits shall not exceed 15,000 tons except for the Black Sea States, which may send through the straits capital ships of tonnage greater than that on condition that these vessels pass through the straits singly, escorted by not more than two destroyers.40 In addition to the tonnage restriction above, the specified forces shall not comprise more than nine vessels during one passage.41
This is how Ulyanovsk would easily get around the alleged "ban" on large carrier transits: it was never banned in the first place. The USSR and Turkey were never belligerents and there was never a Black Sea conflict in the time of the USSR besides WW2. Naturally, these provisions apply even worse to NATO members.
You cannot assume a ship's material condition based on age alone. For example, Georgia BN was in much better material condition than Kentucky was, even though Kentucky was 10 years newer. Kentucky had suffered from an immense amount of maintenance that was deferred until she arrived on the west coast once the decision was made, so was always a week late leaving refit between patrols and screwing over one or two other boats to cover the time she was broken. (This was over 20 years ago, I doubt anyone will care now)
Kuznetsov has about 30 years of deferred maintenance I guess then, since she was only able to be serviced in Mykolaiv where she was built. PD-50 was more of a stop gap until the Northern Fleet could finish their shipyards, which were never finished, because the USSR died. The current "plan" is to rapidly finish this by hastily converting a pair of submarine docks into one fit for an aircraft carrier with a wrecking ball I guess, since Mykolaiv is Ukrainian now, and regardless doesn't have shipyards anymore.
For Russia's purposes Kuznetsov might be closer to 50 years of life.
A carrier that size is basically incapable of doing anything. You need 6-8 fighters minimum, to keep a 24/7 CAP. You need 3-5 fixed-wing AEW or 5-7 helicopter AEW. You need at least 4 ASW helos to have one up and hunting, 5-7 would be better. And you need 2-3 SAR helos to fly while you're doing flight operations. That's 16 aircraft, minimum, just to protect the ship.
It can't launch a strike anywhere, or even put up more than 2 fighters to chase opposing MPAs away.
Better tell the English with their Invincible or the Japanese with their Izumo they're useless before they build three each.
Chasing MPAs away and bombing submarines is how America and Britain won the Western Front.
"Aircraft carriers" are expressly prohibited from the Straits under the Montreaux Convention. Which is why all old Soviet carriers had heavy AShMs and were claimed as "heavy aircraft-carrying cruisers".
But the tonnage limit in the Convention is only 45k. Turkey likes keeping the Convention in effect because it is more restrictive than the UNCLOS. But I cannot see the Convention being renegotiated to increase the tonnage limit.
They aren't, actually.
Turkey just says this because Turkey wants to flex it can stop a USN CVBG with a wall of text. Shtorm 23000E would not be banned even if it were 120,000 tons and armed with a single AK-630, while Nimitz with Tomahawk missiles and SM-2s would still be banned, because Turkey values its relations with Russia more than it does its relations with America and always has.
The privilege of being a riparian state is that you can openly defy the puny navies that do not occupy the Black Sea. It is ultimately Turkey's decision whether they want to even apply the Montreaux Convention and for pretty much all of history of the USSR they allowed the USSR to transit submarines, cruisers, and aircraft carriers that all exceeded the size limits of the Montreaux Convention without penalty (15,000 tons), because they were not at war and they did not want to force a war.
During 2008 there was some discussion in the 6th Fleet of sending the CVN in as a show of force and for disaster relief (they can do the ship to shore power thing IIRC) to Poti, Georgia. What actually happened was a wider discussion with Turkey and they agreed that the 6th Fleet could send the Blue Ridge instead for Operation Assured Deliverance. Russia protested this, because quite literally the Blue Ridge LCCs violated the letter of the law, since the 2008 Georgia conflict was not considered a war and the Blue Ridge was too big, but Turkey allowed it anyway.
As it stands both Ukraine and Russia are under Article 19 of the Montreaux Convention which precludes their transit of armed vessels unless they are returning to their home ports or they are rendering assistance (as the Mount Whitney was) and that only applies during wartime. During peacetime, the limit of displacement of riparian states' ships transiting the straits is functionally unlimited provided they tell Turkey they're transiting a week ahead of time.
The reason for the anti-ship missiles is more the Soviet Navy being somewhat autistic than any insane legalese conspiracy. Ulyanovsk would have been the first proper Soviet attack carrier and Turkey would have no real reason to stop them from transiting despite grossly exceeding the tonnage for a warship. Sea control in Russia simply means being able to leverage multi-axis TOT with anti-ship cruise missiles rather than manned CVWs. In that sense, the P-700s were an economic measure, not a legal one, because it means a dozen fewer tactical fighters need to be carried. Fewer ordnance and jet fuel. Smaller ship for the same punch.
Just mind the launch silos' doors.
Russians design things to function, the way they understand things need to function.
But deck elevators block the functions of a flight deck. I can only assume that the designers had no clue how a flight deck works.
The recovery lane is unbothered and has literally zero obstructions. The deck elevators interfere less with flight ops than the port side elevator on a USN carrier tbh. Ultimately the choice for deck edge vs flight deck elevator is for battle damage over bomb farm size. Evidently, the ship is meant to take a beating and doesn't care much for having a large bomb farm.
Perhaps it's meant solely to deliver tactical nuclear weapons and maintain a small DLI pair.