Excellent prose, Al! What a summary. Carry on!
Multi-classing to get max ranks in a skill is....a poor decision, at best
Arguable. It is perfectly rational for a stealth-based fighter for take a level in ranger, and this is an added incentive. However, the other view is of course that characters who was multiclassed in any case have essentially gotten twice as many skill points with regard to stealth. For popular archetypes such as the rogue/wizard and the fighter/rogue this is a substantial boost, particularly in the latter case.
Paying for cross-class skills each level is not a munchkin's choice, I assure you
Under the new system, of course, the payment for cross-class skills would be equivalent to the payment for class skills under the current one, i.e. 2 skill points to advance Stealth per ranks (vis-a-vis paying 1 x 2 for Hide and Move Silently).
Look at it this way: currently, a person who wants to play the classic thief archetype needs to have:
Does this not have an internal contradiction with your earlier arguments? You successfully argued, and I indeed conceded this point, that the +1sp is a trivial difference in terms of creating 'super-rogues' who are talented in every sphere. The effective addition of +1sp merely enables them to take Escape Artist. You do later admit that "just a bit more wiggle-room" is the upshot- surely just awarding +1sp achieve this just as well, if there really is a problem (which, personally, I don't see).
We've essentially done that. There are really two House Rules here. The first is to combine Hide & Move Silently into Sneak (more on that below answering your post). The second is combining Listen, Scent, Blindsense & Tremorsense and any other non-visual sense into the Sense skill in such a way that any creature with at least one of those gets a chance to use the skill, while those with more than one are rewarded appropriately for having multiple senses to bring to bear on the opposed check (and none of them are now automatic within a certain distance).
The second part I see no problem with, and whilst I am not going to adopt it for my campaign, I would argue is co-equal with the RAW in terms of ease of use, balance and flavour. It's the first which I have the problem with: and note that most of the supposed problems with the RAW are solved by the latter.
That's 4 chances for you to get caught (as a very low roll on either of yours or a very high roll on either of theirs is usually enough to foil the attempt). And that's even taking into account the massive discrepancy in skill bonuses a dedicated stealth PC has over schmucks; dealing with someone even somewhat competent in Spot or Listen and even average rolls can lead to failure.
Ah, clear blue water

. Fundamentally, I think that this evinces a different opinion on the likelihood of the success of stealth at the higher levels. The discrepancy between the stealth PC and the schmucks is, indeed, phenomenal, and by the mid-to-high levels the advantages over one with a cross-class skill are also discernible.
However, the arguments sound good on paper, but numerically speaking, a 10th level rogue with 22 Dex (16 base + 2 level + 4 item), maximum Hide and Move Silently ranks and +5 stealth items has a +24 to those skills, easily enough to beat schmucks 100% of the time under the RAW. Against cross-class skill buyer, with a +2 Wisdom modifier (reasonable- only clerics and possibly paladins are going to have a significantly higher modifier), with his total +8, the rogue's chances of victory are still huge: just over
97% . It is only against a dedicated 'detection' characters with max class skill ranks and 18 Wisdom (realistically, only a ranger) that the rogue is likely to be detected, and the rogue still has a 64.8% chance of victory. The stealth-based character is heavily favoured, particularly by mid-to-high levels when stealth-boosting items are common.
Under the revised system, the %s are the following: 100%, 96.25% (ironically, the +2 synergy bonus means that the detector actually gets a marginal advantage) and 74%. In real terms, therefore, the effect is insignificant. The change is therefore unnecessary.
Re: familiars and animal companions. Not only does Scent only work within 30', but only one character is warned. And in any case, it is better to fold Scent into a separate set of House Rules, without tinkering with Hide and Move Silently.
Detailed analysis of the numerical breakdown shows that, and I was marginally surprised with the results, the practical %s of detection remain roughly constant, though there is a slight advantage under this system against opponents with similar detection skills to your stealth skills. However, the major impact is halving the cost of stealth, and whilst this makes only a slight difference to the skill-point-heavy rogue, to purchasers of cross-class skills, particularly fighters and wizards with relatively few expendable skill points, it is hugely attractive - to the point where multiclassing becomes actively more attractive to get max class ranks. For already multiclassed characters, this is a phenomenal boon.
Essentially, the nub of this question is: is stealth too difficult under the current system? I'd say clearly not. The prevalence of stealth-boosting items and spells, the tendency to have a higher Dex than the detector's Wis and the notion that many opponents will be clearly inferior due to being bound with cross-class skills (which they may not even buy), means that stealth is already powerful enough under the RAW. The change is therefore unnecessary, although there is room for altering the mechanics of the "automatic detectors".