The timestamp can be off a bit[1], it makes no sense to wait for broadcasting as this would only increase the likelyhood of someone else finding a block that also meets the target requirement. If we have two contesting blocks, yours might get orphaned, thus its important to broadcast it as soon as possible.
I think it takes some time until the block shows on all node and one the whole blockchain. I'm not a blockchain expert though so it might be because of an other reason it ook so long to show on the blockchain.
Yes, it takes a bit, but nowhere near 30 minutes.
My understanding is that miners (or mining pools) have the ability to set their own timestamps.
The Bitcoin Protocol has encoded an allowable error of +/- in time, to a max of a 52 hour error.
This is so since it is not possible to have different miners systems all having the correct time, every time. Some lose time and etc.
The block was found recently and is in its proper order. It is just the broadcasted time-stamp (which the miners system says) that is off.
The timestamp is an easy and fast way to manipulate the block in a way that influences the hash, thus it is often manipulated on purpose in order to find a hash that matches the target criteria.
[1] >median(last 11 blocks) and <network adjusted time + 2 hours
https://en.bitcoin.it/wiki/Block_timestamp