Here is the raw data. Basically, I tallied up the number of features innate to a champion's kit for the entire roster, accounting for things like CC, mobility, sustain, manalessness, etc. Keep in mind that these are only the features innate to a champion's kit, without accounting for items, masteries, summoner spells, etc., so all of this is being drastically underestimated relative to the state of those features in-game.

I invite you to look at the data and see for yourself, but here are some conclusions I think can be drawn from it:

• The average champion has about 2 instances of innate crowd control: Typically this crowd control is one hard (i.e. a stun, root, suspension, etc.) and one soft (i.e. a slow, blind, silence, and so on), and the hard CC is more likely than not to be irreducible by Tenacity, as more than half of all hard CC is immune to Tenacity. This includes displacement and knockups, but also suppresses and stasis, less frequently. Effectively, complaints about Tenacity not being as effective as it should be are pretty justified, as just over 30% of all CC in champion kits is Tenacity-immune.
• The average champion has between 1 and 2 ways of increasing their mobility: The most common form of mobility is movement speed boosts, but dashes, both targeted and free (targeted dashes require a unit to dash to, free-targeted dashes can be cast anywhere) come in very close, and together are more common than movespeed boosts. Even the least innately mobile classes in the game, i.e. marksmen and mages, generally have at least 1 mobility boost, and instant gapclosers make up over 56% of all mobility in champion kits. This isn't counting Flash, by the way, which is another instant blink on practically every champion, so there is actually a lot of mobility available to most champions. Those who do not have that same mobility are serious exceptions, rather than the norm.
• Over 54% of manaless champions also have self-healing: By contrast, a little under 21% of champions capable of self-healing are manaless, as most champions with healing are supports and there are no manaless supports. As there are 12 champions with manaless self-healing, a little under 9% of the total roster is a manaless champion with self-healing, and just under 16% of all champions are manaless. There is effectively a decent, though not terribly enormous portion of champions not bound by standard resource constraints, including a fair amount of champions who are liable to be far too survivable early on if their kit isn't gated properly. As of now, 32.5% of all fighters are manaless, 65% of fighters have self-healing, and so 20% of fighters are manaless and can self-heal. This corresponds to many of the complaints players make about top lane, where a great deal many champions can hold on for very long without being affected as harshly by resource constraints as most other laners.

There are likely many more interesting conclusions to be drawn from the above data, but I think the bottom line here is that there is actually quite a fair bit of statistical evidence to back up observations and criticisms players have made about certain trends in the game: there really is a saturation of crowd control and mobility in champion kits, especially hard CC and gapclosers, to such a degree that having at least one instance of either is the norm, and anything below that is considered a deficiency compared to the rest of the roster.

The TL;DR conclusion I also want to draw from this is that there is a lot of feature creep in League of Legends: when each new champion design is released with either multiple forms of CC and mobility, or some other special feature to compensate for the lack of them, in order to remain competitive with the rest, it leads to a lot of repetition across kits, and to designs that feel more cookie-cutter than they ought to be. Not every champion kit has to be a bunch of pure nukes like Annie or Veigar (both of whom incidentally have hard CC), but when a champion needs to have mobility and/or CC in their kit just to be able to compete in an environment that is overflowing with both, doing so raises the issue of it contributing to a larger problem of game-wide feature overload, one that already feels like it's limiting design today, and that risks limiting it even further as more champions get released and updated.