The OP asked which is the strictest agency so GUE and UTD came up. Before anyone jumps on the GUE / UTD band wagon they also need to look at the overall cost vs benefit situation of going that way.
GUE and UTD are both expensive not just in terms of courses but also the style of diving. UTD Tec 1 guys have to use Helium as a standard gas below 100 so if they are doing decompression dive at 120 feet on standard gas, they spend 90-100 USD on back gas alone. TDI divers get to choose for themselves when they need Helium. For decompression diving in the 120 feet range it is uncommon for divers to be using Helium and the cost of the back gas is half for the TDI guy than it is for GUE/UTD fellow. In the long run this means that a man with limited budget will dive more if he stayed away from GUE/UTD even though they train divers to a higher level. Some of my UTD friends feel trapped because firstly you need to log a certain number of dives to progress to the next level and secondly the cost per decompression dive that these guys incur is more than what my TDI dive budies are paying. Standard gasses are expensive for shallower depths and so is the mandatory gear and lights etc.
In another thread, I asked about Cave training and to be full cave certified through a mainstream agency, the training expense is coming to be 1500 - 1800 USD. If I was to do GUE Cave 1 then that course alone is 2600 USD range. Keep in mind that GUE Fundies would have to be cleared as a pre-requisite course which is another 800 - 900 USD. The total cost would then be close to 3500 USD for GUE Cave 1.The cost in which GUE is training me to start my cave diving is the cost in which I would become an experienced cave diver with 170 - 200 real dives after my certification if I went with NACD. The question then should not be if GUE's newly graduated Cave 1 diver is better than an NACD newbie graduate Full Cave diver because they are half the price of each other. The question should be if GUE's Cave 1 diver better than an NACD Cave diver with 170 - 200 real dives under his belt because they are both created in the same money.
Let me make this thread simple for the OP.
If you wish to be a highly trained diver who has no money left to dive because he spent it all in training then GUE / UTD are the way to go. If you wish to be the poorly trained diver who gets to dive a whole lot more because he saved money by sticking to cheaper certifications then PADI,SSI, SDI, NAUI etc are the way to go. Choose how you wish to be miserable and dive.
One of the most important of the UTD tenets is to create a thinking diver. Much like other members have posted, for dives outside of the training environment one is free to adapt the procedures as applicable to the situation. So, if a UTD team decide to do some dives down to 120 ft without helium, that is something that the team needs to consider and decide on. There is no UTD police going to come and take away your card. The flip side of the coin is that cost vs safety is a dangerous area for many. I have friends who tried to save on scrubber costs on a rebreather and found out the hard way that saving 20 bucks isn't worth a life. The economics of diving are unfortunately often a part of the incident chain, from cheap training to bad compressor maintenance to inferior equipment to unserviced LSE, the list goes on.
RAID is worth to look at.
As far as the OP goes, I think the answer to the title question is :
GUE is the most stringent, followed pretty closely by UTD. I am only referring to the equivalent of OW and AOW here, I cannot speak for the cave and overhead environment as I am not there yet. I imagine that the same would apply though.
As far as RAID goes, I do have some comments, again NOT from an OH point of view. I started as a NAUI instructor in 2003, one of the things I loved the most was the "Freedom to Teach" philosophy as well as the "Loved One" standard. This meant that I could teach any number of additional skills, require any passing standard that i wanted to (as long as it met and exceeded the base standards) and even if a student passed and met all my requirements, if they were not someone I would trust to be the buddy for my son or wife I didn't have to pass them.
When I started diving rebreathers, I looked at RAID for my certs for a number of reasons. While I was doing this, I did my instructor crossover as i was really impressed with the online training and the support for the students wrt theory training as well as the basic philosophy.
What I liked especially was the fact that, like GUE and UTD, the training was designed backwards from a very technical skill set, so that I never have to tell a student "For this course you will do it this way, but later on you will do it that way" which I have always hated with a passion.
Things like donating the primary, all skills to be performed on all courses with neutral buoyancy, in the hover, horizontal etc are, for me, basic building blocks for training new divers.
What is also nice is that the standards are in some ways like NAUI of old. For example, a student must show proficiency with a "non-silting propulsion method". So I can teach frog kick and modified flutter etc on an OW course as the students needs dictate.
With GUE and UTD, there can occasionally be issues in remote dive locations when the operators are not familiar with the agencies (I only know PADI syndrome...) so often my UTD students end up dual certified. When I do a UTD Rec course, it meets all the requirements of a RAID OW course and so, if they have done the online theory, I can sign them off with no additional requirements. With PADI, there are several extra (and to me obsolete) things that have to be added in.
On balance, I feel that the commercial agency with the highest requirement for OW skills to pass is RAID (no knowledge of CMAS or BSAC so apologies if Im insulting our friends)
On a last note, AFAIK RAID is the agency that stipulates the most minimum hours in water for certification (6 total of which 2 hours MUST be in OW). This helps to limit the "4 dives of 20 min each" sausage machine that I see often in the PADI-type agencies.