Personally, I wouldn't find that as interesting. Saul and Lizzy are the young couple. Theres still some mystery about their relationship and whats going on (pregnant or not?). Burt on the other hand has served his real purpose to the story I feel. There isn't really any mystery about him at all as he has been pretty open. Your point about us getting to see him as a shattered and falling apart old man would not be anything new, as he has been doing that pretty much the entire series so far (blames himself for tommy/saul/lizzy, can't shoot straight anymore). It may sound harsh, but in my opinion the only way Burt's character can progress is in a heroic death.
Also, I didn't notice any real attraction to Tanya. I'll give it another listen though.
EDIT: I don't think you can really justify that there a relationship between the two of them is hinted. He says he likes her, but that is obviously just a reference to the joke she made about Angel.
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