Your intuition is broadly correct. More dice give you a better chance of a critical roll, regardless of other circumstances
The intuition for this situation is that more non-hunger dice means more chances for getting at least two to show tens. As long as that's the metric that matters (meaning we don't care about results that are not tens), then adding more dice always improves your odds of rolling additional tens.
This is completely unrelated to the hunger die (I'm assuming that a hunger die is a specific die marked out before the roll). The results of the hunger die roll are not related to the results of the other dice, and so for this particular scenario it may make things easier to think about the two varieties of dice completely separately (it may be easiest to imagine rolling each set, hunger and non-hunger, separately):
- If the hunger die is rolled first and shows a 10, more non-hunger dice increases the odds
that at least one of them will also show a 10. This matches your
- If the non-hunger dice are rolled first, more dice means greater odds
that at least one of them shows a ten. Rolling the hunger die
afterwards then gives a 1/10 likelihood (or 1:9 odds) that the hunger die
will show a ten, yielding a messy critical.
It might be a bit confusing to think in terms of assuming a result (although that is the correct way to calculate the odds you're interested in), because you are removing the randomness of the assumed result. That seems to me to be the cause of confusion in your second-to-last line.
A master at lockpicking is more likely to roll a critical by virtue of their large dice pool-- that's the representation of their lockpicking skills. But their hunger die has an invariant chance of showing a ten. They are very good at picking locks, but have a chance of their hunger complicating the effort, and are not necessarily particularly good at dealing with that hunger just because they pick locks well.*
The rough math
The easiest way I can think of to calculate the chances of a messy critical for a given dice pool is first to calculate the chances of at least one die showing a ten. The easiest way to calculate that is to compute the complementary event (no die shows a ten).
For d10s, this is just (9/10)^N, where N is the number of dice you are rolling. We are computing the 90% probability that each die will show a non-10 number, and multiplying those probabilities to get the likelihood that those outcomes will all happen at once.
With the number of rolls that produce zero tens calculated, we subtract that probability from 1 (here, 1 represents "100%", the set of all possible outcomes). One minus the probability of no ten results equals the probability of one or more tens.
So, with our chances of rolling at least one ten from the regular dice sorted, we now look at our chances of rolling a ten on the hunger die. This is always 1/10 for a single, fair d10. That makes the math pretty easy! Whatever the probability of rolling at least one ten from the regular dice, we have an additional 1/10 chance of a ten on the hunger die. That means that your probability of a messy crit is just 0.1 * [probability of at least one ten from the regular dice].
This does, indeed, increase the chances of a messy crit as dice pools get larger.
*Narration always matters
Sometimes in TTRPGs mechanics alone make for odd-seeming results in narrative terms. I'm not familiar with V5 rules overall, but I suggest that, narratively, a messy crit of "ripping the door of its hinges" during a lockpicking check really is unreasonable. It's hard for me to believe that anything done with a lockpick and torque wrench, by anyone or anything, would lead to the door being torn apart.
I don't have a good understanding of what a critical success in picking a lock would be... it's a binary event, you succeed or you don't, unless we factor in elements like how long it takes.
Instead, a messy crit (in the spirit of a botched roll from Revised or V20, which may not be appropriate) might more plausibly lead to something like jamming a lock pin permanently out of place in the lock cylinder. Maybe it makes a loud noise and attracts attention. Certainly the lock is obviously broken, and no key will ever work in it again.
Conversely, if the hunger die represents the character going out of control (similar to a frenzy in earlier editions), it might represent the character losing control and becoming so impatient with picking the lock that they lash out and tear the door off its hinges.
If there is a mechanism for things like automatic successes, a character with a large dice pool for a check might be a good candidate for one due to this messy crit mechanic. If you don't need, or can't use, a critical success, then there's little point risking a messy crit.
That said, Vampire has never been known for being really technically airtight and well-balanced. This may be another example of that, carried forward into the latest edition!