Does the word Lego have anything to do with Latin verb lego? When I started learning Latin and learnt the verb "legō" specifically (meaning, "I collect, gather"), I suddenly realized "Oh, so that's where the Lego name comes from!" and it made perfect sense.

It was somewhat surprising for me to learn later that in fact it originates from Danish "leg godt", meaning "play well". I actually still believe Lego has relations with the Latin verb, even though there is no clear proof of it in official sources.

Do you think Latin "legō" has anything to do here, or is it just a mere coincidence?
The official story is that Ole Kirk Kristiansen didn't realise until years after naming the company what it meant in Latin (normally it's said to mean "I put together", not "I collect"), and as there was no construction over the products made in the first many years, it seems unlikely that he should have thought along those lines.