The origin of "welcome" seems to be "will come", not "well come". This is clear from the German version "Willkommen", which is closer to the original proto-germanic "wiljakumo" and is directly related to the rather dated German expression "Will [er/sie] kommen!" ("Let him/her in, or "Will he/she come [in]!"), or even to English "will ya come!". The connection of "will" and "come" signals the desire to receive the guest.
The french "bienvenue" seems to be a too-literal translation from wiljakumo [0]. A similar construction is missing in Latin.
Take "welcome."[1] Well come. You've arrived well. You make a good entrance? Not a ridiculous thing to say when someone arrives.
What about in response to "Thank you?" You are welcome.[2] Huh? What does that mean?
Or, "you are welcome to take a second cookie." How does that translate into "I do not mind if you take a second cookie?"
[1] Welcome translates quite literally in other European languages. For example, in French, it is bienvenue - bien: well - venu: came.
[2] On the other hand, it is much more common to say de rien (in French) or de nada (in Spanish): literally, "it's nothing."