It is not arbitrary*. It's called "judgement".
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*You probably aren't doing this, but I've seen it several times recently - misuse of the term arbitrary. Arbitrary means, "because I feel like it, and I cannot give a rational justification". If you can back up with some logic as to why the thing applies, you are not being arbitrary.
Let's not start arguing about the language, because I am not an English-native speaker, and you can make a fool of me anytime because of that.
But in case you want to know, by "arbitrary" I did mean that as a DM I feel I have to make a non-objective (or sometimes metagaming) decision. So yes, I am guilty!
Not sure what case you have in mind. First of all, I admit I'm having in mind myself
designing an adventure. Maybe you have in mind yourself
using an existing adventure, and thinking about all the challenges and obstacles as "already there". Thus, if the PCs end up in front of a locked door and want to pick the lock, it's understandable that you feel the DC for lockpicking is
objective, and you are just making an objective decision (or actually no decision at all!). It's not your fault if the lock is impossible, and neither if success is automatic. This case wouldn't really make me feel bad, whatever the outcome, because I do not have to make that arbitrary decision.
OTOH, if I am designing an adventure OR if the adventure says there's a lock but doesn't provide a DC, (there are always guidelines in the rules system, but ultimately the DM has to pick a number), then I very much feel like I have to make an arbitrary decision! If the system allows the PCs to Take20, I already know what DC they
will beat. So I am stuck there, having to make a decision between "they will succeed" or "they will fail".
This is a kind of decision that as a DM I do not like to make. It feels to me the same as having the PC in a conversation with NPC, and having to decide if they "win" or "lose". If they play badly I can make the
subjective decision that they "lose", and viceversa. But more often than not, it's a close call, and I want uncertainty to be solved by a dice roll rather than my decision, otherwise I feel like I'm being "deux ex-machina" or playing "mother may I" too often.
My problem with Take20 is that it takes that option away from me. If Take20 is in the system, I cannot say "you still have to make a check" here. If I do, then I do not need the Take20 rule in my game.
The GM is expected to apply it every once in a while. The GM decides loads of times that a die roll isn't even called for - when tasks are insignificant. So, that decision's being made implicitly dozens of times a game, and this ceases to be a solid argument against Take 20.
I can still apply that if Take20 is not available. But you say so yourself, "when tasks are insignificant". I am concerned with significant tasks, and that's the DC range at which Take20 matters.
Take 20 can be used to deal with, "You can try and retry this pretty much ad infinitum. If you've got enough skill and ability, you *will* get it eventually, if you don't, you won't, so let's not take up the time of everyone at the table while you try it." Because sitting around for someone to roll dozens of times to see what happpens if and when a 20 comes up is generally *BORING* for everyone else at the table.
Yes, it was clear to me since the early 3e days, that this was the motivation.
Still, Take20 is not the only way to avoid boring rerolls.
No retries works just as well.
There is also a feeling of metagaming that bothers me a little here. The idea that a PC
would retry 20 times something until successful, comes from the fact that the
player knows a dice is rolled, and "early or late" it
will yield the maximum result. That's not really how things work in life.
At least, if the penalties for Take20 would mean something (e.g. someone here suggested a very interesting system of increasing times), then we can discuss it. But for those skills I have problems with they don't, because 20 times 1 round equals 2 minutes, which is an irrelevant amount of time outside of combat.
