At no point in the story does the Dovahkiin join the Blades; he/she is never given that option. Joining them would involve taking the same oath and being issued the same gear that your followers get if/when they are recruited into the organization. That does not happen and the possibility of it happening never even comes up. The Blades are not a joinable faction in Skyrim.
Now you can RP that you joined them if you want, but that's your choice and yours alone. My Dovahkiin never joined the Blades; my Dovahkiin had no interest in the Blades and didn't know they still existed until one of them decided to force her way into my business.
Did she do that so she could decide if I was worthy to join their special little club? Apparently not, because I was never asked to join. On the contrary, I was asked to go where they wouldn't or couldn't go, obtain what they couldn't or wouldn't go get for themselves, and do what they could not or would do on their own. I was not a member, I was just a very powerful and unique and useful tool. I don't see how that obligates me to them in any way whatsoever. I didn't take an oath to serve their goals or do their bidding. I served those goals insofar as they coincided with mine, and I did their bidding when it seemed like the best or only way available to get where I needed to go next. In that respect, I treated them exactly the way they treated me. It's not my fault that they still need me to do things for them after I've outgrown any need for their support and assistance.
