Your proposed study would almost certainly cost time and money and then be dismissed with no change in plans. At worst, it could turn into a political argument that would entirely kill the Sheppard LRT for another generation - since the plan for that is to have it tunnel down and terminate alongside the subway, and they'd have to entirely stop that until the study had been chewed over.
Yes, I know the LRT is on hold right now... probably so that the BD-SSE argument can wheeze along until the next election for political brownie points. However, in general, the paperwork is mostly ready, and if they would make some firm decisions and commit some funding, the Sheppard LRT could break ground right as the Finch LRT finishes up.
Since I'm bored at home on painkillers with a sore back... here's my armchair assessment of your suggestion with things you may not have considered, and some back-of-the-napkin revisions of your numbers. Where I don't know exact numbers, I've tried to lean a little in favour of your proposal. I hope you won't think me rude for shooting it down, this is intended to be an honest appraisal, but it's just.... there's absolutely no way it's going to happen.
Let's start with your estimate of time saved. Preliminary planning of the Relief Line north is already saying 2 minutes at peak and 3 off-peak. So the per-person, whole day average station dwell is maybe 1.25min, and 1 minute taking an escalator and walking at a properly designed interchange for an average rider. However, given what's in that area and the current station box, your proposal necessitates a shepherd's-crook tunnel (no pun intended; see the pic attached below) which would add at least half a kilometer of track to the total distance, and it would make at least 1km of it slow-speed, thus eating up a couple minutes or so with extra driving compared to a straight shot. Overall, the full day average per-person-per-trip savings would be 30 seconds at *best*, and at worst possibly a net loss depending on engineering requirements and resultant speed limits, not the 3-5 minute savings you guessed.
Assuming the best, how many people will save that time? Well, first of all, it's almost the same identical distance for Don Mills to Queen by way of the Relief Line as it is to go via Sheppard and Yonge. The big factor in this is the time; if the break-even time point is east of Leslie, then not one person will have a shorter trip available to them by implementing your plan.
However, let's assume fewer stations on the RL, so it operates at a faster average speed than the Yonge and Sheppard Subways. The Shep had a ridership of 48k in 2015. Of the 48k, 30k passengers can be lopped off the top; that's the bare minimum number of people going between Don Mills and Shep/Yonge, who will either not use the Shep at all once the RL opens, or are travelling east-west anyways and are thus unaffected either way. I think it's fair to say the number uncertainty caused by people going short hops between Bayview and Leslie cancels out the small number of people going from North York Center to say, the Science Center. The other three stations combined total 18k people of traffic per day. How many are then going to Queen/downtown, as opposed to the rest of the city? The SSE study concluded 23% of Scarborough riders are going downtown, so let's assume it's just a little higher at 33% for Sheppard since there's more transit infrastructure. Besides, it makes for nice round numbers.... about 2k passengers from Leslie, 1k from Bessarion, and 3k from Bayview - some of whom will take the RL regardless for comfort, some of whom will take Yonge for direct routing no matter what, rather than walk a few blocks from Queen, or transfer back onto the YUS downtown. So again, it depends where the break-even point is. I can't see it being at or west of Bayview, so while there will be a bunch of folks heading to the RL from there, that's for comfort only, and the time-savings argument doesn't count. Bessarion, Leslie... it could be faster going either way. We won't know for a long time, until stations on the RL are decided. No matter where the break-even is though, your 30k number is mathematically impossible, I think it's more realistic to estimate 3-4k people, at thirty seconds' time savings each.
Your resulting cost is a huge low-ball. You guessed $100M total. That's to add another half kilometer of curved track under residences and a park, that's more complicated than a straight shot with an interchange. That's if it's even possible with the highways. The STC subway extension is $3.3B for 6km of fairly straight track and a new station, so call it $250M just for the extra tunnel and track - probably more. As a result of doing so, they'd also have to rewrite the LRT plans, most probably by widening the Sheppard bridge and having the LRT terminate at ground level with a transfer; lengthening the transfer time for more passengers than would benefit from the interline. Regardless of if it's done middle-of-road, or on the side, some cars would have to cross LRT tracks to get on to the highway at some point with this plan, unless they converted Sheppard Ave from a partial cloverleaf to a full. Alternatives: making a second bridge entirely for the LRT, or tunneling double-deep below the highway and subway and probably below the waterline. (Can't be between the subway and the surface, there's not enough room.) Any way you slice it, it's a big chunk of extra money, definitely in the tens of millions, possibly over a hundred mil. Add productivity losses from Sheppard Ave and highway lane restrictions, probably an extended Don Mills station closure, new environmental assessments, an extra emergency exit somewhere along the curve, and probably extra noise baffles to satisfy justifiably angry local residents. It'd more likely be somewhere from $500M-$1B, rather than $100M capital investment after all's said and done.
On top of the capital, there's the ongoing costs. Wasted capacity and overservicing alone would eat up more money than even the maximum possible time savings are worth. People consider the Shep a ghost town today, and after the Eglinton Crosstown opens, plus the relief line - interlined or not - a goodly half of the current Shep passengers will change their route. The majority of the remaining passengers would use it as a 1-2 stop hop. Heck, they could probably shorten the Shep down to two-car sets, though keeping it at 4 would be better to maintain transfer capacity in case of a subway emergency. They might shut it down on weekends, or kill it completely and convert it to LRT right through (and yes, I know that'd require seriously painful design work, but it just might be less of a fiscal loss at that point.) Meanwhile, a Sheppard-to-Downtown RL will likely have numbers comparable with the University-Spadina arm, running twice as often as the Shep runs right now, and they're building it to be expandable to six-car trains. Even with short turning most trains and the resultant headaches that would create, that's four cars worth of waste per train on the Shep at all times. Not to mention that future expansion northbound would be permanently impossible without major work, pretty much guaranteeing there would never be a subway link to Seneca, nor to Markham/Unionville, at least not in the current century.
In short... hundreds of millions of dollars up front, a net loss just in passenger minutes, plus greater day-to-day operating expenses for the TTC, and no future-proofing. It's simply not a sound idea.