"What crime-ridden city/country are you visiting where day bags are being snatched from under tables with the owner present?" It can happen in a city/country where such a thing is totally unexpected.
Picture this. My work colleague went to Europe annually for many years. As a classical music lover and performer, her trips were always focused on that. Most of her trips were small group tours with music lovers like her. She was one of such a group seated at a large round table for dinner. She'd put her day bag, actually a large purse, under the table at her feet. I think they were in Austria or Germany. I know it wasn't the UK, Italy or France.
When the event was over, she discovered that her bag had disappeared. Hers was the only one taken. I don’t think it had all her valuables in it, but it did have her room key. When the group got back to their hotel, the management let her into her room. Her bag was there. So was her room key. The cash was gone, but her meds and credit cards were still there. To her, the biggest loss was the many music CDs she'd already purchased on the trip. All of them had been stolen from her room.
I didn’t return to European travel myself until several years after this incident, but I remembered it well. I'm not a person who carries a large purse anywhere. My MO when I lived in Germany and when I traveled before that was a small cross-body purse plus a mesh bag or plastic bags from local vendors. On my husband's first trip with me in 2009, he questioned me about that, wondering why I wasn't concerned about the fact that people could see what was in those bags. I told him that it was highly unlikely that anyone would grab a bag if they could see the bottled water, bread, cheese and fruit in it.
Like others have mentioned, in a restaurant I slide the small purse down around my waist, even here in the US. It may end up in my lap, under a thigh, or next to me, but then only on the wall side of my body if there is one. On planes or in enclosed booth seats, I may slide it around to my back and use it as a lumbar support.
So my answer is that I don’t use a day pack. Of the contents you listed, a little pack of tissues is the only thing that I'd be carrying in such a bag and those fit nicely in my new small and comfy cross-body purse.