That depends on where you live, no? I have moved to an area that is as flat as the Netherlands. Even my 10–33 cassette is overkill. Unless you need sub-1:1 gearing a lot, I think you can easily make it work, especially with a 1x13 drivetrain.
It is true that you should think about gearing and make your own choices rather than accepting defaults.
What is the terrain like where you live? What gearing do you currently have and want/need?
Regarding moving the groupset over: in principle yes, but really look at the new frame. Can it take a 2x chainring? Does the BB work with your cranks? Are the brake mounts the same? I had this plan on my last bike build, and ended up replacing/changing most parts anyway. You might be better off buying a new bike with an OEM-priced groupset, and keeping the old bike intact to sell or ride.
On my 9-speed bike I had very sloppy shifting running 11-25 with a MTB derailleur (& the 12-21 corncob was a complete non-starter), not because of cable pull (cable pull ratios in the 9-sp world are the same between MTB & road), but because the parallelogram angle dropped the guide pulley even further away from the next sprocket with each click of the lever. This is because the RD was designed for 3-4t jumps between sprockets, not 1-2t jumps.
This is just a word of caution for @shasell if they want to run vastly different gearing between two sets of wheels on the same bike. I wouldn’t expect the shifting to be nice on a bike with an 11-28 cassette & a MTB or gravel derailleur… unless if it’s somehow no longer an issue in the 11-13-speed groupsets. Having to swap out derailleurs each time too because they’re matched to the cassettes, & then also the chains because the RD cage lengths are different, that sounds like a headache, a recipe for offending the neighbors with my potty mouth. 
I settled on a Cervelo Aspero as my road/gravel bike, w/ a chunky wheelset from Light Bicycle, as well as a 45mm front/55mm rear aero wheelset also from LB. I’m smaller, so the 48/31 gearing works fine for me on road. I named it “Gus”…because…Aspero Gus…
Ha. I do the same dual wheelset thing with my Crux, Æon Crux
This is a good suggestion. Even if some components are not top-notch, you can upgrade it over time.
I just watched @DesFit’s products-of-the-year review, and he got an Aspero as his do-it-all bike. He claims to be very happy with it and doesn’t feel the need to get “proper” road bike. When I will need to replace my aero road bike, it’ll definitely be something like the Cervelo Aspero or 3T Exploro.
Depends where you live. I’m in a fairly populated bit of UK where it’s nice to get offroad and away from the traffic, there are some great trails that aren’t too technical, but it’s rare to go more than a few miles without needing to do a bit of riding on quiet roads to get to the next offroad section. A typical longer ride will involve at least 30% and maybe 40-50% on tarmac. I don’t need a MTB for the trails I’m doing, and a gravel bike makes the road sections much quicker and more enjoyable. Plus a lot of the quieter roads around here are pretty rough at times - gravel, rough surface and potholes, wet leaves are a pretty much constant feature in winter. Gravel bike is a nice hybrid option for mixing it up on crappy roads and light offroad! An “all road” bike with ~40mm clearance would also work but honestly I think all these bike categories overlap with each other now, it’s just semantics/marketing.