Without magic-derived healing--meaning, spell slots, openly magical features like Lay on Hands, and inherently magical consumables like healing potions--how can the martial characters keep up with the quantity of damage they receive from combats?
Because you only get back half your HD with each Long Rest. Meaning, in the absence of magic-derived healing, you get one day of full-ish healing, and from there on out you're simply not going to recover properly unless you take a full day doing nothing dangerous so you can get back all of your HD. I've been the party healer (indeed, a party healer as a Celestial Warlock, so my cures were always fat!), in a group where no one else was a spellcaster. I'm quite well aware that HD simply don't keep up with the damage output of even 5ish combats, to say nothing of the 6-8 or beyond that folks seem to think martial characters can do without a sweat.
Because the martial healing I mentioned isn't all based on hit dice and hit dice recovery during a long rest is irrelevant in a discussion where long rests aren't applicable. The parties start the day with full everything. They cannot start a long rest until at least 16 hours after the previous long rest. They're taking short rests. Short rests recover some spell slots once for wizards and some sorc points once for sorcerers. Short rests recover a rage, a second wind, and all focus points.
In that 16 hours to start another long rest and then 8 hours to complete the long rest the all martial group can take many short rests even if long rests become part of the conversation.
The comparison was between a party with a barbarian, fighter, monk, and rogue vs cleric, druid, sorcerer, and wizard at 10th level in a 10th level adventure. Looking at HD that you mentioned that's 10d12, 10d10, 10d8, and 10d8 hit dice for self-healing for the martial group; and the fighter and barbarian likely have 16 CON while the monk and rogue likely have 14 CON. That's ~310 hp of HD healing. The caster party has 10d8, 10d8, 10d6, and 10d6; and it's possible the Cleric and Druid have 16 CON while it's more likely the sorcerer and wizard have 14 CON going for DEX instead IME. Benefit of the doubt favoring the casters all having 16 CON would give the casters ~280hp of self healing that way, running -30hp deficit even with that benefit of the doubt.
The 10th level fighter (champion) has 4 second winds that heal ~62hp and the fighter recovers 1 second wind every short rest for 15.5hp each or possibly for other uses instead but given the heroic inspiration available the likelihood of that need is low. But the fighter can take a short rest and heal that 15.5hp regardless many times if needed.
The barbarian (world tree) has 4 rages. Each rage grants the barbarian 10 temp hp (40) and the start of each of the barbarian's turns lets the barbarian grant 3d6 (~10.5) temp hp to an ally near him. Rage lasts up to 10 minutes but combats don't. 3 rounds of combat is about 31 thp, and using bonus actions after the fighting is over to continue raging allows for topping up to a higher average leading into the next combats. The upper limit is 190 thp total but the realistic number is ~60-70 between what's the barbarian's 10 thp and what he hands out. That's per rage so ~240 thp plus ~60 per additional rages recovered on short rests.
The barbarian also only takes half damage on most typical attacks while raging.
The monk (way of mercy) has 10 focus points that he can fully renew once per day with Uncanny Metabolism and also renew every short rest. Uncanny metabolism also heals the monk ~14.5 hp in the process. Mercy monks can replace 1 attack with their Hand of Healing for free when they use Flurry of Blows. IE the Nick monk spending 1 ki does 2 attacks with extra attack, 1 off hand attack, 2 attacks with flurry of blow, and one healing touch or a 3rd attack with flurry. Assuming 20 DEX and 16 WIS for this build, Hand of Healing heals ~7.5 hp at that level so spending maybe 7 focus points out of 20 before short rests that day would be ~52.5 hp of healing. Maybe 3 uses for each short rest afterwards for ~22.5 per short rest. This leaves a lot of focus points for other uses.
The monk also has evasion at no resource cost, and deflect attacks with unlimited uses on reactions. A 20 DEX monk deflects ~20 damage per use with the ability. A monk deflecting 2 attacks in 8 encounters prevents ~320hp of damage.
The assassin doesn't add healing. Rogues are just good at avoiding damage with Cunning Action, Cunning Strike, Uncanny Dodge, Evasion, and in the assassin's case possible use of Roving Aim via kiting.
How many short rests can this group take in the same amount of time it takes the casters to take a long rest? I'm not going to get into that because it turns into some huge numbers but I wanted to make that point. 2 or 3 short rests in pretty easy to accomplish, however. So the HD disparity is ~30hp, That's ~700hp of healing and thp before defensive abilities that prevent damage including the 30 hp deficit. With HD healing that's over 1000hp the martial party has in a day without pushing it.
Spell healing was so bad 2024 rules had to buff it. You can show the spell healing, thp from wild shape, and channel divinity and we'll compare further. Then I'll keep on track and point out that each spell used for healing or temp hp is a spell not used for damage (or control or versatility).
"A long way" doesn't mean "all the way." Which is the key problem.
Except skills don't run out and can be used from start until finish, but spell slots run out and cannot go all the way, which is the day point. A lot of spell options when slots runs out becomes an illusion of choice.