Although the accuracy in the old days wasn't what it can be now, I think that part of the entry-level tone issue is due to the instruments being mass-produced. Go back in time to admittedly more expensive instruments & hand building and you'll find that the luthiers were selecting the tonewoods themselves, rather than just buying job lots of suitable cheap tonewoods like Agathis. If the raw material's not as good, then the accuracy of the dimensions won't really count for much.
Spyke wrote
I'll vouch for the Squier name. You're not going to pick up the first one you see and it's the one you buy. You will need to try a few to find a winner. I had my heart set on a white one when I was shopping but of the 7/8 that were available there was only one white one and the one of the black ones felt best. I bought the black one.The electronics are entry level at best but that can be tailored to the student's specific needs / flavour.
Spyke's made some good points, which I honestly think you can apply to nearly every make under the 25-35 grand price tag. There are good ones and bad ones - well maybe not bad, but just not "good" - from all manufacturers of a reasonable standard. I played about 10 Les Pauls in four different shops on Denmark Street in London one day (I had bad G.A.S. at the time). They covered brand new Studios, Standards and Customs to a '75 Custom. The pick of them? An '81 Custom and an '80 Deluxe, and the Custom wasn't 700 quid better than the Deluxe - which was a heavily worn plain red (not my fave LP colour) axe, but had the sweetest voice. I'll also venture that half of the new ones would have struck me differently had they been better set-up. Set-up (imo) makes the difference over this price range, together with that indefinable quality that makes a guitar "right" for you and "wrong" for someone else.
But back to entry level.
Choices really are limited in terms of good BEGINNER guitars. I think you can find some good stuff if you're willing to tear yourself away from the name on the headstock, i.e brand snobbery. We all want to have Stratocaster/LesPaul/Telecaster/Ibanez on the headstock like our idols when we're beginning, but a lot of people limit themselves to Squire and Epiphone - or cling to them like a drowning man with a lifebelt. They all make some lovely beginner's instruments, but you need to do a lot of sifting sometimes.
In this price range the Yamaha Pacificas shouldn't be overlooked by someone after a Strat-type guitar, especially the 112 (HSS). Guitarist ran an article a little while back where they modded 4 on a budget, and got some very interesting results. They're good to play as well, and definitely better than the Bullet-series Squires.
With Gibson being so full of it & Epiphone unavailable new, two options are the Vintage V100 and Hamer XT-Series Sunburst Archtop (actually a lot like a double-cut Les Paul). The Vintage is a very close copy of the basic LP to the uninitiated (and fools at a distance), and they're good guitars to play in this bracket. They're new to our market but have been well-received in the UK for some time now. The Hamer's a very nice guitar in this range and well worth a look.
There's also the 2nd hand market. I picked up a mint (and I mean mint) Squire Standard Tele in vintage sunburst from a guy who was really into accoustic and wanted to just shift it. It had a Graphtek nut fitted already and just needed a bit of fret dressing (the 18th was a bit high & choking notes on bends). Tweaked the pickup heights to improve the sound a bit. Bought that for R1350.00, the new price at the time was around R3400.00.

I've since passed it on to a friend's kid who has started learning electric. There are usually loads of barely touched Squire/Epi/Ibanez RGs on Gumtree for sale by kids who thought one strum made you a hero (usually with practice amps and a Boss MT2 included!). Worth a look.
I will agree that the electrics are really a big let down. There's a reason why they're not as good as higher-priced offerings in the same way that an Indica's not as good as a BMW. My gripe is that things like pots have been around for decades, so by now we shouldn't drastically lose tone when rolling off the volume (like a new Hamer semi-hollow Epihone ES-335 competitor I tried recently did).
I still think that set-up will help a lot in this bracket, but stuff like pickup upgrades on a bulet Squire or Ibby RG aren't really worth it. You're better off saving the money and then spending it on an overall better guitar as your playing improves.