Quick solution. Tie it in with android or their Xperia phones.

You have great points :), but handheld gaming is kinda a past thing now, the only reason it's still selling becuz of exclusives, same with Console ofc. I think the next PS5 or Xbox will be just an app like Steam but not just for PC. For TV, Smartphone & Tablet as well. Look up PlayStation.Now, the Cloud gaming and VR might be the next big thing for gaming.

It's a good thing btw. Since you no longer have to spent hundred of dollars to buy a console or upgrade your PC to match the requirement of the game or be able to play their exclusives like Halo, Uncharted. You only need an HDTV dat has "PS5" or xbox cloud gaming app supported, a PS/Xbox controller and a stable internet connection, then you're good to go. Prices of the games would drop significantly thanks to everything become digitally, so they don't need to pay fee for manufactures, shipment and retails. I expect the game will be priced at cheap as $39 for an AAA game at launch! Another perk is that you can playing it where ever you want since like i said, it have smartphone & tablet supported.

Unfortunately, Sony is a really big company and doesn't rely on 1 source of income, so they probably don't see the need to make another handheld for now, at least

The handheld you just described would be far more expensive than the Vita already was.

I don't think they'll bother with another handheld... The only reason I think it might continue to work for Nintendo (though at lesser sales than even now) are Pokemon/Monster Hunter/Fire Emblem etc.