Hussar
Legend
I have not found it so. For example, a Fighter can on average withstand one hit per level he has attained before being killed. So a 3rd level Fighter can only expect to survive 2-4 hits from goblins or orcs before expiring. Fairly common monsters such as ogres do large amounts of damage (1d6+2 in OD&D, where hp are 1d6 per level for Fighters, or 1d10 in Classic, where Fighter hp are 1d8). Healing is limited, resting can be problematic... it's not hard to get killed even at fairly high levels.
I've never played OD&D, so I cannot comment on it.
But, in Basic/Expert D&D, you have plate mail at first level (60 gp, pretty easy buy), meaning that you've likely got a 2 AC at first level, before any Dex bonus. That ogre is only hitting you about 30% of the time, and orcs and gobbo's are down to about 10%. Sure, you can only take a couple of hits, but, you don't take too many most of the time. And, beyond 3rd, you're pretty much God, because, you can take 1 hit/level, and the baddies hit you so rarely.
I know that IME, combat damage death was a FAR second place to Save or Die death. To the point where very, very few PC's died to hit point attrition, but, lots of PC's died to snake bites.

The Secondary Skills table is great.
Look at a 'modern' game that has the same sensibility: Over the Edge. In OTE, you can have stats like "Surgeon: 3 dice" or "Ex-Green Beret: 4 dice". What do those stats cover? Whatever a Surgeon or an Ex-Green Beret should be able to do.
The Secondary Skills table is the same way. You were a smith? OK, so you can shoe the horses and construct a forge. You were a tanner? Well, you know a lot about p***. And so on.
Though I don't actually need the table. Just write down your background. That's what you know how to do. Why do I need rules for that?
Funnily enough, this is what I've done with 3e Profession skills. If you can make a decent argument for why your Profession skill (whatever it may be) applies to the current situation, I'll allow you to use Profession in place of whatever skill might be called for. It works for my bunch.