Like ctuser1, I am concerned about the dynamic. Your wife doesn't "let" you be a father; you are a father, and you have just as much right -- and responsibility -- to care for the child, to take her out, to change diapers and gives bath, etc., as your wife. So to the extent you are not doing those things, you are the one "letting" her dictate the kind of father you will be. So if you don't want to be that kind of father, you cannot let your wife limit your involvement -- and that means that when she objects/says no, you need to stick to your guns, and quietly, calmly, insist that you will do X.
I absolutely agree with the others about the anxiety issue, and you need to find a way to get her to therapy -- even if it's couples therapy and the only way she goes is you drag her with you kicking and screaming (figuratively, of course!). But I also wonder if she might "hear" you better if you approached the discussion from a slightly different angle. When I read your list, it sounded like me-me-me -- I want to go have more fun, I want to do this, I want to do that. I know, I know, this is your venting, and that's all perfectly normal and fine, and I know what you mean by it all. But I wonder if your wife got that sense, sort of an undertone, even if you were trying to be really really patient and kind and undemanding. If I were your wife, and you approached me with "why can't we go out like we used to, or let's plan a weekend away, and boy I'd really like more sex," my immediate, knee-jerk response would be "goddammit, I'm busting my ass already trying to be a mom and hold down a job and take care of the house and the cooking and all that, and I'm barely keeping my head above water as is, and now he needs me to be Julie your Cruise Director and plan dates and trips, and then somehow pretend I have energy at the end of the week to have sex?" And that would either make me angry, or if I were dealing with anxiety and PPD, just remind me of what a failure I am at everything. IOW, if the discussion comes across as you wanting to have more fun -- even with her -- that's going to be perceived as just another obligation, when she is already feeling at the end of her rope with the obligations she already has. I will tell you, the hardest times in my marriage were when I felt like I had three kids instead of two, because my DH was being a doink about something. And that did not put me in the kind of mood to want to go out on a date and have sex, you know?
What about if you approached it from the role of father and husband, i.e., focusing on your daughter's and wife's needs instead of your own? Try: you want to be an active father, you think it's important that little girls have strong relationships with their fathers, and so you want to spend more time with her doing the daily, normal stuff and be able to build that special relationship with her, and when your wife won't let you do something as simple as run a bath or take the baby on a walk around the neighborhood in a stroller, that really hurts you and makes you feel like she doesn't trust you to do the most basic things. That you see how hard she is working to take care of the baby and the house and the job, and you're worried she's running herself ragged, and that's not fair to her, and you want to help share that load so she has time to take care of herself. Maybe ask her for three things she'd like you to take over with the baby, or three things she'd like to do for herself if you guys can make time for it, and see what she says.
Beyond that, I'd advocate lowering your expectations and taking a win where you can get it. There's absolutely no reason you guys can't take the baby to the park for a few minutes, or go for a walk/short hike -- those things that you're asking to do as a family are extremely reasonable and the stuff that life-with-kids is built on. So set a goal of getting out to do something very simple, low-stress, and not time-critical like that every weekend. Once your wife sees that the world didn't end, maybe you can push it to a trip to the grocery store or to church. (And then when your daughter has a meltdown and your wife freaks out, just stay calm and take care of things and refuse to treat it as an excuse never to do that again, because all kids do that and you managed it just fine).
And finally, do see a therapist yourself for tips on living with someone with that kind of anxiety/PPD.