Yes
No your translation calls it obsolete.
The actual text says being old, growing old is near disappearing.
You are saying more than what is stated.
Saying the destruction of the Temple has something to do with this ignores the fact that the Covenant changed upon the resurrection and Jesus' assent.
I'll quote the NASB since it's a bit more literal (these are not "my translations") but I'll highlight the word in protest since NASB also uses obsolete
Heb 8:13
When He said, “A new covenant,” He has made the first [obsolete/old]. But whatever is becoming [obsolete/old] and growing old is about to disappear.
the verb root is παλαιόω = to make old, declare obsolete, age, wear out. the text uses 2 different forms however.
the latter is "becoming-old" which seems to be the definition you are drawn to
the former is πεπαλαίωκεν or "He-has-made-old" [the first].
-παλαίω- is the stem, which means something old or ancient (and you agree with)
the prefix however πε- is the perfect tense showing a completed action. so πε-παλαίω- carries a meaning of a completed old.
-κ- is another perfect tense marker
-εν is the third person maker
so together carries a meaning of he has made old, he has rendered obsolete, he has worn out. The interesting part is it does not age by itself, but it is a completed action of making old performed on it by Christ. Christ has made old the covenant. Obsolete is a bit specific more abstract word, and a very English way of putting it, but once you break down the word, it's what the meaning is in practice.
I speak a different language (Melayu), and it translates it saying in the sense of "already make old/outdated. Some translations use the word for old some use the word of outdated. Both carry the sense this is a completed action already performed on the object.
I'm happy to leave it more literal "he has made old" but I fail to see how that changes the meaning since it's not old because of years gone by, but it is old because Christ made it old, in this sense it is ordained old, with the new being presented along side of it also ordained new. Obsolete/replacement are images that naturally come to mind but I get if you perhaps have an allergy towards "abolish" language. To be clear, Jesus is not abolishing, he is making old, but he also makes new and shows a better way.
This reminds me of the language in Rom 7:6 "we have been released from the law so that we serve in the new way of the Spirit" again an aggregated word and loaded "we-have-been-released" (κατηργήθημεν) and like Heb 8:13 we do not do the releasing. We were made free from law by Christ, Christ is the one doing the releasing and he is the one also that makes it old. "so that we serve in the new way of the Spirit"