Only talking about iOS (but I suppose it's similar for Android): There is a huge number of icon sizes. Some are only relevant to older devices that you might not support anymore, so you leave those out.
If your designers create each icon pixel by pixel, then you have a problem. It's also very, very inefficient. Normally they would design icons as some vector graphics, and then it is very easy to convert that vector graphics to bitmaps in all the sizes that you want.
They then need to check whether the small sizes look alright. Your icon might have details that look nice on a big icon or one drawn on a high resolution screen, but don't work at all in low resolution because the user cannot figure out what that detail means. In that case you would create one more vector graphics where tiny details are removed and the icon simplified if necessary, and use that to create the smaller icons. Of course both icons should be as similar to each other as possible, because the user might use your app on two different devices.
If you worry about the sizes of icons or other graphics, make sure that you don't get graphics with all kind of metadata included. I once received 200 icons, buttons, tiny images that were about 50 Kbyte each, total 10 MB. A free tool removing metadata reduced most to less than one Kbyte without any quality loss.
And in principle you should try to avoid a first bad impression. Icons are the first thing a user sees. If they say "that app is rubbish" before they even launch it, that's bad and should be avoided.