Yes....the fact is that the Jews were commanded NOT to adopt
anything that has pagan roots, especially those connected to spiritism, which was most of them.
Deuteronomy 18:9-12...
“When you have entered into the land that Jehovah your God is giving you, you must not learn to imitate the detestable practices of those nations. 10 There should not be found in you anyone who makes his son or his daughter pass through the fire, anyone who employs divination, anyone practicing magic, anyone who looks for omens, a sorcerer, 11 anyone binding others with a spell, anyone who consults a spirit medium or a fortune-teller, or anyone who inquires of the dead. 12 For whoever does these things is detestable to Jehovah, and on account of these detestable practices Jehovah your God is driving them away from before you."
All those things were "detestable" to God, so we need to look carefully at what we are doing by way of such things that
seem like harmless celebrations, because the things that the Canaanites were practicing had their roots in satanically inspired "spiritism" which is the devil's work.
As you mentioned, we see no birthdays celebrated in the Bible and no birth dates ever recorded (including Jesus') because the Jews did not celebrate their birthdays. The anniversary of the day they were born was not observed,
because it was a pagan practice. The birth date was used to make horoscopes in order to predict a child's future. The cake with candles (originally tapers) was used to ward of evil spirits as were the "wishes" directed to the one celebrating the birthday. Nothing changes but the attitude. God still hates it, no matter how much we justify it. It makes kids greedy just like all the other pagan celebrations....adopted under a different name.
Only two birthday celebrations are mentioned in the Bible.....Pharaoh's and Herod's.....neither of whom were worshippers of the true God. And someone lost their life on each occasion at the whim of a proud ruler.
If you think it doesn't matter, read Luke 16:10.