Cut yourself some slack OP.
three is still pretty young. Some kids that age can play independently, some can’t. I’ve had one that could and one that couldn’t.
Lots of TV - we can get into this habit too. It’s a habit as much for me as a parent as much as it is for them. Happens more when I’m run down or overworked. Eg before Xmas TV became a default for our DC (now 5&10) including eg during dinner.
I’ve learned that I can reset but I have to be rested and resourced to do so. So although I didn’t like it I let it slide up to Xmas as I was crawling there myself. And then through Xmas as we had a lot of relative visiting to get through. And then after Xmas while I recovered a bit. THEN the last weekend of the holidays I addressed it.
changing TV. When I’m ready I go cold turkey on TV. Otherwise I find each day is about waiting for the TV hour. Going cold turkey (we did three days) is a tough first day of the five year old especially asking and asking but then he accepts it’s not happening and is more likely to engage with other stuff.
after a few days of none at all it’s easier to bring it back with better boundaries (eg we do 1hr before dinner)
i agree with suggestions to do with them what you also enjoy.
im also trying to structure time together - so to be clear with them and myself when I’m focusing on them and available to play (their choice) or available to do an activity I’m organising (eg craft or baking which I quite enjoy).
and then when I’m doing my stuff and they can either help me out or entertain themselves. I do play with the kids a lot but am also aware some time together can become poor quality as I’m distracted by things I need to do or waiting to get some time for my stuff.
ifs about finding a balance of playing with them, supporting them to develop the skills to play themselves, and (while I agree with a PP that it’s sad if your parent didn’t ever play or do stuff with you) also age appropriately letting them know that you are not a 24/7 entertainment machine.
but somE three year olds aren’t yet able to entertain themselves for very long at all.
OP parenting a young child can be draining. Find a way to take some time for yourself to recover and THEN consider what changes you want to make