I can't tell by what you have said about these tees if the sentimental attachment is because it reminds you of a special time or special person or special place - or if the sentimental attachment is just . . . because. (which is OK!)
I have a couple of tees that I have held onto because of sentimental reasons: one is from a family reunion and one is from a race that I won when I ran my best time. I wear them only when at home - usually lounging in the evening. Otherwise they stay tucked away on a closet shelf and because they don't take up much room, I am fine with holding onto them.
I also have a couple of tees that I wear when working with the animals at the shelter. I learned the hard way, that little fur-balls can be really hard on clothing - and so finally decided that I needed to stop wearing anything but tees and old jeans when working there.
FWIW - The style of tee that you are showing in your pics is not really considered a modern silhouette - but I don't know if that matters to you. In addition, many women think of tees as more like an under-shirt, rather than an actual top - and I think that the older we get, the more true that becomes!
So - I think it's really a personal decision whether or not you hold onto those tees. Since you said that they are looking worn and tired though, I wouldn't wear them out in public! BUT - just to be clear, I wouldn't wear anything out in public that is worn and tired. (Also, if you want a fun tee with fireworks on it - like the Lucky one - there are plenty out there that you could find to replace it!)