[RPG] How necessary is mechanical character advancement

Really, there are more than two kinds.

Open Fully point based, à la Hero System or GURPS: anything you can justify to the GM, you can (eventually) get. The problem is that such systems often overwhelm players and GM's with options, and often, some options may be mechanically useful but out of character, and taken anyway for the mechanical benefits alone. A few use multiple pools of points in character generation, such as EABA uses, but not in play.

Restricted points systems, Like WFRP, or Dark Heresy and Rogue Trader, or the old GW Judge Dredd, leave lots of choice to the player still, but focus it by limiting the choices. They usually allow much more expensive options for outside skills. Your choice of class career focuses the choices to manageable ones; it also often puts some that might be better fits for character development into higher expense, or even off limits. Some implementations attempt to balance, others don't. WFRP1, for example, had some careers with 5 advances, and some with more than 20, at 100XP per advance. Many have randomized generations

Class & level based skills like Palladium's games, or The Arcanum. Your class determines your skill list; your character level determines what those skills are rated at. Most allow some elective skills; some, by spending experience (like the Arcanum), others at specific levels. Limited choices, characters often feel carbon copy. Fast, though, if using character sheets with the skill list preprinted save for levels.

Class based skill points per level as in D20, Alternity, Rolemaster. Your class determines what skills cost what, but generally, all skills are available to all classes, save a few iconic ones. D20 adds a second category, Feats, which are also restricted, but less so. These systems allow quite a lot of freedom, but still focus the choices by expense. By using an experience system and levels, it also sets a rate of advancement limit, so skill based tasks have a maximum possible range. Guarantees that any character who loses focus suffers compared to those who don't, but also allows characters who want a key skill or two off class-concept to get them... at a price.

Classless Level Systems a very few games have no classes, but still use character levels. Flying Buffalo's Mercenaries, Spies, and Private Eyes has skills, but they are rasied by "Use it enough"; character level raises attributes in MSPE. A couple others simply tell you to advance a set number of skills when you level up, or a few attribute points.

Some of these actually have classes, but the class only determines special abilities, and those don't change with level; level determines attribute or skill advancement. The best example of this is FGU/Goblinoid Games' Starships and Spacemen.

Use it to have a chance to it as in Chaosium's BRP and Pendragon, FFE's MegaTraveller or Mark Miller's Traveller. These systems, you mark what you use. At end of session, season, year or whatever increment the designer (or GM) chose, you make a roll to see if it goes up. Almost always requires tracking instruction hours to acquire new skills, almost always limited solely to gaining skills, not other abilities. Reshapes the character to be good at what the party has been doing, requires plenty of bookkeeping, hard on character sheets. Pendragon uses years, BRP session, MegaTraveller months, and T4 a limited number per session. The rate of rolls has a strong influence on the sense of both completion and advancement.

Use it enough and it goes up as with MSPE, Mouse Guard or Burning Wheel. Each skill tracks experience separately. When enough counted uses accrue, it goes up. MSPE, it's a set XP per use, and when the skill levels up (using the same XP track as character level), raise it. Burning Wheel requires a number of easy uses, a number that are close to ability, and a few that are "guaranteed" to fail but tried anyway. Mouse Guard requires successes and failures. The various methods of earning the experience make the games play very differently. They all share an accounting nightmare, namely, XP tracks for every skill. The good news is that advancement is near guaranteed; the disadvantage is that some players will work the system to get the needed difficulty. Mouseguard has the problem of mandating failures, which was the #1 player complaint when I ran it.

Class for starting skills, points thereafter A few games use template style character generation, but it's open points thereafter. Classic Battletech RPG 2E is the primary example, with Cyberpunk 2020 being the second one, but d6 system (including WEG Star Wars), and Shadowrun also allow it (but don't require it). It makes for fast, carbon copy character generation, but with the ability to add what you need later.

Class for special abilities only just like one or another of the above, but each character is in a class that either allows a special skill (as in Cyberpunk 2020), or a specific special ability subset (as in WoD/Storyteller). The advantage is that you usually get suggested skills, but are not stuck to those. The drawback is that the special abilities often drive character concepts in certain directions, and bucking those can be a problem.

And I know I've not hit all the bases yet, just the broad brush strokes of most of them.

You use the best progression from your two classes. Player's Handbook, "Multi-CLass Benefits and Restrictions", page 44, last paragraph before the multi-classing example:

If the optional proficiency system is used, the character […] gains new proficiency slots at the fastest of the given rates.

which is reiterated on page 52, end of the first full paragraph:

Multi-class characters can use the most beneficial line on Table 34 to determine their initial proficiencies and when they gain new proficiencies.

You apply this progression as you gain levels, which ends up meaning that it applies to both classes separately. Your Cleric 8 / Thief 9 should have gained 1/4 × 8 = 2 weapon proficiencies† and 1/3 × 8 = 2 NWPs (rounded down) from cleric advancement since 1st level; plus 1/4 × 9 = 2 weapon proficiencies (rounded down) and 1/3 × 9 = 3 NWPs from thief advancement since 1st level; for a total of 4 WPs and 5 NWPs gained since 1st level.

A common interpretation of the multi-classing rules is that you must fill each weapon and non-weapon proficiency slot according to the restrictions of the class level that earned it; consult with your DM to see how they handle this. Regardless of that, see the notes on multi-class priest weapon usage restrictions on page 45 of the Player's Handbook before you choose any thief weapon proficiencies that you might not be able to use.

† Though one might expect this should be multiplied by level − 1 (not the full number of levels) since you already get stuff at 1st level, the rules are more generous and grant new proficiencies at each level divisible by the rate. See PHB p. 50, "# Levels" (centre column, 3rd paragraph).