The Skilled feat lets you take 4 Skill or Tool proficiency, regardless of if they're on your class list or not. Sorry I don't run hackfests. Players know that going in and if they build themselves a severely skill-limited character, they'll find themselves having difficult participating in a number of situations. It's not like they didn't know when we started the game. Want more proficiency? Roll a Variant Human and take the Skilled feat. Or make a half-elf.
I apologize for being pedantic, but technically the Skilled feat gives 3 proficiencies, not 4.
People have TONS of choice in 5E. Class+Background is typically 4 skills and at least one tool proficiency. Race usually beings one to two more. That's 4-5 skills for the "low skill" classes, usually two more for the "high skill" classes which roughly covers 1/3rd of the skills available on the sheet. Take the Skilled feat and now you've got nearly half the skills available covered.
While it's true that most backgrounds give
some kind of tool proficiency, the vast majority are effectively useless: instruments, gaming set, or artisan's tools, none of which have any practical application to most adventurers' lives. (Anecdotal, but, I can say I've had *one* situation where a gaming set prof might be useful...IF it's the appropriate game, which is highly unlikely; I've never had a situation an artisan's tool would be a "make or break" difference.) Others are going to be campaign-specific; not much use to Navigator's Tools or Vehicles (Water), as they won't make much of a difference if you adventure on land. Disguise Kit, Thieves' Tools, Herbalism Kit, Poisoner's Kit, or Forgery Kit would all be useful a plausible amount of the time...but not one of them is an Artisan's Tool, so the vast majority of backgrounds can't get them unless specified.
As for getting half the skills covered? Well, I guess. There are 18 skills. If you're willing to blow your first feat on skills, and willing to restrict yourself to a single race choice, then sure, you can get 2 (class) + 2 (bg) + 2 (race) + 3 (feat) = 9 skills, which is exactly half. I dunno where you get that race "usually" brings in another skill--only Variant Human, Half-Elf, Elf, and Half-Orc get any bonus proficiencies, and given how many DMs I've seen who don't allow Variant Human because they consider it broken/overpowered/etc., that's really only 3 out of 9 races. Certainly not a "usual" thing to get AN extra skill, let alone two.
Sorry but there is MORE than enough choice and plenty of ways to go about getting more skills. Noone has any excuse for being unskilled beyond A: they want to be, B: they don't know how. I'm happy to educate the latter and I've got no problem with a person who wants to be unskilled, provided they're not making that choice in ignorance.
Unfortunately, you're characterizing "wants to play something that isn't some kind of Elf or a Variant Human" and "wants to play something that isn't a Rogue or a Bard" as "wanting" to not be skilled, which...well, okay you can do that, but it comes across kind of rude. E.g., someone who wants to play a Dragonborn Paladin, and who thinks trading ASIs for feats is a raw deal (in most cases), "wants" to be unskilled? Uh...no. I'm keenly aware that that limits you to a narrow set of skills in 5e, and I find it a little frustrating. (To be honest, I find it a little frustrating in 4e as well, and it doesn't even have the "Proficiency is hardly noticeable at level 1" problem, nor the "give up precious ASIs for feats" problem.)
Edit: Presuming a DM allows a player to take the Skilled feat more than once, and allows multiclassing, it is actually possible to get proficiency in all skills (and Thieves' Tools, too). Half-Elf Lore Bard/Knowledge Cleric 1/Rogue 1 with two applications of the Skilled feat gets 6 (feats) + 3 (Bard) + 3 (College of Lore) + 2 (bg) + 2 (race) + 2 (Knowledge domain) + 1 (Rogue MC) = 19 skills, meaning you can pick up a useful Tool proficiency you didn't get from BG/Rogue MC (presumably Herbalism Kit, but Disguise or Poisoner's Kit wouldn't be bad either), and you get Medium armor prof and Shields to boot. If you max out the rest of your levels as a Lore Bard, you'll also end up with 8 skills, or 7 and Thieves' Tools, Expertised (4 from Bard, 2 from Knowledge domain, 2 from Rogue), meaning you're proficient with every skill AND an "expert" in nearly half of them. Since you do end up with 1 "excess" skill/tool prof, you
could play an Elf, Half-Orc, or Variant Human, the last of which having the substantial benefit of making up for the ASI you lose for not reaching Bard 19--but, contrariwise, you pay for it up front since the Half-Elf gets +2 Cha *and* +1 to two stats of choice, meaning in the end the Half-Elf is arguably more "efficient" (starts out 1 skill behind but with better stats, ends with same stats but 1 tool prof ahead).
Yes, I have seriously thought about playing this character, should I ever find myself invited to a new 5e game. This is not a likely event, mind, since I don't much care for 5e as a system, but I figure it's best to have a character concept ready should the opportunity arise (much like how I want to play a
Dragonborn Dragonic/Dragonspawn Monk, should I ever get invited to a 13A game).