Wouldn't fix anything expense wise.

Back a few years ago, the space where we keep our dryer and washer, plus other things, needed some work done. Since it was only a few weeks, we just moved them to the screened screened in porch.

If anything, the gas bill went up a little bit, though that could have just been from the door opening more often to the porch. Power bill stayed the same for this time of year.

It isn't like a dryer vents enough inside air to matter at bill time compared to having to run longer to heat the air up that little bit longer and/or take longer to get things dry.

If you want to save money, hang more stuff, save the dryer for things that can't/won't dry fast enough to prevent any issues like mildew (happens some in cold weather here, and a lot during really humid days). Thick blankets, heavy jackets, sometimes denim. Or when weather doesn't allow air drying in bulk.

You can rig a short line inside for bad days, delicates, and such. Depends on how much space you have as to how much you can hang though.

Besides, aren't dryer vent covers and hoses standard sizes? You'd have to deal with that as well, which is secondary to having to extend the intake and cut a second hole of there isn't an efficient way to combine them.

You'd be better off trying out one of those indoor vent exhausts, where you just plug the vent hose into what amounts to a filter that keeps the lint from flying everywhere. It achieves the goal of not "wasting" already heated air, and it'll even use the vented air as heat for wherever the dryer is. You do have to deal with humidity, but winter air tends to be too dry anyway. Positioning the exhaust device well can limit the issues from the humidity.

There's even dryer heat exchangers that slightly reduce energy used, and can even be connected to heating for floors or other systems (though I've only ever seen one hooked up to an under floor system.

And, if you fuck around with the dryer, you might be able to rig something yourself. But you'll likely discover that it's cramped quarters. Which is part of the reason you don't often find them connected externally. The fan units are crammed in as they can, and may or may not have an opening in the case of the dryer. You'd likely need to cut wherever on the casing it is, then build your own hose attachment, then run hose to wherever you want to intake air (the only time I've seen this done, it was to pull air from attics, where there's either air heated by the sun, or slightly warmer air from indoor heating seeping through). But it does require some degree of fucking around.

Trying to manufacture dryers with it by default would make them bigger, more expensive, and not gain any real savings unless it's coupled with other methods.