Cardio is NOT for losing weight. In fact in many cases it actually is very difficult to burn fat while doing lots of cardio because it trains your body to burn the fact more efficiently. This is why you'll see people spend so much time at the gym and get no results. The OP is quite correct that without a serious change in diet coupled with the cardio you will NOT lose weight. Which is exactly why you shouldn't do cardio if you want to lose weight, you may get a small drop in the beginning, but ultimately you will return to a baseline. Pushups etc. are great for warmups, but ultimately these will have very diminishing returns as well (the "100 pushup" style of training is laughable).
If you want to burn fat and lose weight you really need to lift weights, and you need to lift heavy weights. The idea that you can "tone" your muscles with small weights is just ludicrously wrong. You wont turn into a mega-hulk, trust me.
If you want to increase your stamina, your lung capacity and your heart health cardio is absolutely what you should be doing. But also understand that this comes at a cost of diminishing your muscle gains (thus slowing down how quickly you burn your fat).
TLDR: the best way to lose weight is to lift heavy weights in low reps (as heavy as forces you below 10 reps) with moderately short breaks between exercises. Cardio will be an uphill battle to lose weight.
And the best part is you actually get to eat MORE if you are seriously lifting.
Edit: if you want to disagree with me and downvote, at least post a response. I'd love to hear why you think I'm wrong.