Lets say we have 2 programs constantly generating random 53 bit numbers. First 32 bits are just system random and last 21 bits are the last bits of current timestamp. The programs probably don't have the same current timestamp, but also the generated numbers aren't compared at the time of generation. Does adding timestamp decrease probability of collision?
00000000000000000000000000000000 000000000000000000000
random timestamp
Appending the timestamp to a 32 bit random number can never lead to more collisions. In the worst case, the timestamp part will always be the same and the chance of collisions stays as high/low as before.
But overwriting 21 out of 53 bits (= 40%) of random data with something like a timestamp can drastically increase the collision chance. Whether that is relevant or not depends on ...
Using pure random data without incorporating time will result in an equal or lower collision chance, unless the timestamps are so coarse and the clocks are so out of sync, that the timestamps rarely overlap. But in that case you are effectively using the time part not as time, but as a fingerprint unique to the system. For something like that there are better approaches. Here are some examples, even though none of them are perfect:
Using fingerprints like these is a gamble. Overall, it can decrease the probability of collisions. But in the case of a fingerprint collision (even if it happens very rarely), the collision chance is drastically higher as long as the fingerprints stay that way. Therefore, I'd recommend to use only randomness. It is easier to implement and collision probabilities are evenly distribute across all cases, no matter under what circumstances your systems run.