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It is common to thank somebody for "a kind invitation". However, when somebody writes
I kindly invite you to event xyz,
does that not suggest that the person who invites the other is doing the invited person a favour? If yes, the meaning would be quite different from phrases such as
I cordially invite you ...
or
It is a pleasure to invite you ...
My google search produced many hits for the phrase "kindly invite", but all of them point to websites of non-native speakers/institutions.
The meaning of "to kindly invite" is straightforward. It means what you think it means. You are also right in thinking that it is different from the other alternatives. It appears rude, as the inviter is suggesting that he is doing them a favour. For this reason, it is uncommon to write that you are "kindly inviting" someone, the other two examples are far more common, with "cordially" being preferred.
The use of kindly in this manner - kindly invite, kindly request and similar - is a feature of Indian English.
Although very common on the subcontinent, its use elsewhere is almost unheard of. The other phrases that you suggest would be more common in other parts of the world.
