Do Rogues become Tier 3 by removing restrictions on sneak attack?

I enjoy playing Rogues, but they generally have a hard time keeping pace in a 3.5 game and are only Tier 4* per this answer. I’ve personally found one of the biggest hindrances to Rogues is the absurd number of creatures flat out immune to sneak attacks. Indeed of the 12 general types of creatures, 5 are completely immune to sneak attacks, which includes things like Undead and Constructs, which are often walking piles of hit points and usually plentiful in many campaigns. Not to mention the number of creatures of Huge size and greater whom can reasonably be construed as having unreachable vitals by the Rogue under normal conditions.

The ability’s description is as follows:

If a rogue can catch an opponent when he is unable to defend himself effectively from her attack, she can strike a vital spot for extra damage.

The rogue’s attack deals extra damage any time her target would be denied a Dexterity bonus to AC (whether the target actually has a Dexterity bonus or not), or when the rogue flanks her target. This extra damage is 1d6 at 1st level, and it increases by 1d6 every two rogue levels thereafter. Should the rogue score a critical hit with a sneak attack, this extra damage is not multiplied.

Ranged attacks can count as sneak attacks only if the target is within 30 feet.

With a sap (blackjack) or an unarmed strike, a rogue can make a sneak attack that deals nonlethal damage instead of lethal damage. She cannot use a weapon that deals lethal damage to deal nonlethal damage in a sneak attack, not even with the usual -4 penalty.

A rogue can sneak attack only living creatures with discernible anatomies—undead, constructs, oozes, plants, and incorporeal creatures lack vital areas to attack. Any creature that is immune to critical hits is not vulnerable to sneak attacks. The rogue must be able to see the target well enough to pick out a vital spot and must be able to reach such a spot. A rogue cannot sneak attack while striking a creature with concealment or striking the limbs of a creature whose vitals are beyond reach.

With that in mind, if the criteria for sneak attacks was changed to delete the last paragraph, would that move the Rogue from a Tier 4 class to a Tier 3? The definition for each being:

Tier 4

Capable of doing one thing quite well, but often useless when encounters require other areas of expertise, or capable of doing many things to a reasonable degree of competance without truly shining. Rarely has any abilities that can outright handle an encounter unless that encounter plays directly to the class’s main strength. DMs may sometimes need to work to make sure Tier 4s can contribue to an encounter, as their abilities may sometimes leave them useless. Won’t outshine anyone except Tier 6s except in specific circumstances that play to their strengths. Cannot compete effectively with Tier 1s that are played well.

Tier 3

Capable of doing one thing quite well, while still being useful when that one thing is inappropriate, or capable of doing all things, but not as well as classes that specialize in that area. Occasionally has a mechanical ability that can solve an encounter, but this is relatively rare and easy to deal with. Challenging such a character takes some thought from the DM, but isn’t too difficult. Will outshine any Tier 5s in the party much of the time.

I suspect that this has been attempted before at someone’s table before. A good answer would discuss that experience and whether the player felt that Rogues ascended to Tier 3 in usefulness.

*Full discussion on tiers provided here.