A stock 7.3 diesel won't really out pull any newer big gasser. Only get better mileage doing it.
Dumping a bunch of money into an old vehicle to make it marginally better is an act of futility and generally not frugality (see what I did there...?).
Also spending "all" your budget on an old vehicle is a bad decision unless your other budget includes $ for repairs.
All 4 kids are now over 16 it appears, so are they really all going on trips with you? Doesn't seem likely unless they're 16 and quadruplets, lol. Truck may work but those are as expensive or more than comparable SUVs.
Something was wrong with the Excursion or your remembrance of the old burb was skewed. Have had both drivetrains. Early 2000s V10 Triton smokes an early 2000s 5.3 towing.
There are good $10k V10 Excursons out there. You started the same thread asking the same question as the one below this one. Did you read it? But excursions are more rare than good 3/4 on suburbans now. But there's a fair number of low or lowish mile 3/4 burbs out there too. At least up here there is. If you wan the right vehicle you may need to search geographically further away.
BUT after all that, why are you looking for a bigger rig if you were happy with the old 1/2 ton Burb save for its power on the grapevine and similar?
1/2 ton Tahoe/Yukon and Burbs are literally dime a dozen. Not hard to find a low mile older one. And if you step up to the late 2000s you can get a 6.0 or 6.2 6 speed burb 1/2 or 3/4 or Denali which will either be heavy duty or have 400hp if you get the 6.2.
The something else is a van or a pickup. You won't get the power out of a van for the most part you could out of a Denali. And you will be driving a van the rest of the year when you're not towing. Real question is, do you want to look like a contractor/creeper in a van or a soccer mom, going to the grocery store?
2016 Ram 2500, MotorOps.ca EFIlive tuned, 5โ turbo back, 6" lift on 37s
2017 Heartland Torque T29 - Sold.
Couple of Arctic Fox TCs - Sold