Honestly looking at the weight you can handle now I don't think the problem is so much strength but more that pull ups just feel awkward. There is no reason you shouldn't be able to do even one pull up unless there is a past shoulder injury.

A band would definitely help. We have those pull up/dip stations that have the weight assist (a little platform you put your knees on) but a band works the same way. Also a training partner would help. I find, at least for me, that pull ups are more about the feel than they are about the weight.

I noticed you didn't have any pull downs in your back workout. You may want to add those in as they are basically the same movement but easier to do. If increasing your pull ups is a priority then I would factor those in.

I guess it goes back to the original question you asked. In theory you should be able to grow the lats without pull ups. You seem to have a solid back routine. But I look at it like this: can you grow a bigger chest without benching? Can you grow bigger legs without squats? Some will say yes, some will say no. I think you can but the exercise is really so good that you will hurt the development process by not doing it. I just read something (can't remember where, maybe here) about a bodybuilder who hurt himself doing squats so he switched to higher rep leg presses, ditched squats completely, and he said his legs grew even more than doing squats. The difference here is obviously you feel like you aren't growing the way you should.