Hmm, from my experience people have not been slow to voice complaints about how much the meals at fundraisers cost, or other such things. The mode of giving has never come up though.
I also attend an Antiochian parish, and if it's easy to see what everyone gives then maybe that isn't as covert as it should be. Ours is a basket, not a plate, and nobody can see what you put in besides the person holding the basket.
I think you do have a point on "interrupting" the service to some degree. It's not something I had ever put much thought into, probably because I've never heard anyone complain about it as I said. I'm sure there are better ways and my parish has always been open to adjusting how we do fundraisers and other such giving events.
We do not attach names to all payments, only for the basic giving commitments formal members have made. This is a spiritual matter as much as it is a financial one, and our priest leads by example every time and gives as much or more than anyone else. Surveillance is not the right word to use, but being a visible member of the community is valuable. I do understand privacy concerns though, (especially in this world).
Being pushy certainly isn't the goal, and it is not something I've seen come up as an issue so your point surprised me a bit. I don't think my parish is married to taking collections in the middle of service and would be open to change if there was a need, but we haven't seen the need ourselves.
Also, concerning the verse you quoted in your original post, I don't think it is as much about giving in secret as it is about not being prideful in what you give. Giving secretly can also be useful, especially if it is an amount that could easily lead to pride. Pretty much every large donation that has ever happened at my Church has been done anonymously for this reason. It's a balance!