I also completely agree with Moreslidingdoors
Marital rape and slavery are very different from being given away at a wedding. However women being the property of her father than husband, marital rape, and slavery are all disgusting archaic remnants of a bygone era which pretty much everyone now agrees should never be brought back.
If you agree with the above, and I find it hard to believe anyone wouldn't, then surely you can see that a gentle nostalgic nod towards one of them is a pijt on a spectrum to a gentle nostalgic nod to all of them. And gentle nostalgic nods are steps on the spectrum towards thinking they were nice things.
So at one end of the spectrum we have a bride being walked down the aisle, a father being asked for blessing / permission for hand in marriage, and further along the same spectrum we have women with fewer rights than men, women paid less for the same job, and even further along we have slavery and women being the property of men.
I think it's fair enough to say you don't want to take any steps along this spectrum. If your opinion is that the first couple of steps are on for you but no more then fine - as a PP said: own it. But don't pretend that there is a magic line after walking a bride down the aisle where everything one side is OK and everything the other side isn't. You've drawn that line according to your own preferences, own it and accept others may be valid in feeling differently and worrying that allowing any steps is a slippery slope.
Fwiw these issues are the main reason I'm still unmarried after 15 years together. We are now planning to marry and I am going to have lots of internal battles about what to choose as I'm very close to my lovely dad, but I cannot publicly nod towards the idea of him giving me away.