I have Fralin Vintage Hots in my Strat, which Matt previously used before he went to the Amalfitano's. I can get a nice tone on the bridge pickup (which has a metal baseplate) by doing what you were ie. rolling the tone off a little and maybe reducing the volume slightly too...I found the tone on all 3 pickups to be much better by having them set relatively low - bass side barely protruding from scratch plate on neck and middle with a slight slant upwards to treble side. I have bridge pickup set even on bass & treble just slightly higher than the other two.
I had a set of Fralin Blues Specials complete with Base Plate on Bridge Pickup, to be honest they sounded much the same as the Lollars that I currently have fitted - the neck / middle and "in between" sound great - the bridge is very weedy and thin.
I'm relatively inexperienced with Strats - my other guitar is a Hot Rodded Telecaster (Fender Japan Ritchie Kotzen) this has a mini humbucker in the Bridge position and it absolutely sings - I can roll the tone back and get something that is close enough to the Matt Schofield tone - with a Strat, I can't get near.
My pickups are set quite low - about 1.5 - 2mm protruding from scratch plate - tilted as you described. I learnt the nasty tones that you can get if you raise them up too much. My bridge pickup is a lot closer to the strings to balance out the levels - I'll try dropping everything down a bit to see if that gets me any closer.
I also have a Fender Eric Johnson which has low output pickups - this is no better when it comes to getting the "Fat, singing OD Tone". The temptation is to drop a humbucker in and be done with it but you know how the hunt for tone goes....sometimes you just have to take the most difficult route!
Like many I have found that the stock OTS does not "like" Strats - it's strange because my immediate reaction is "Strats = lower output from pickups = crank the gain" but this really doesn't help in my experience. I have found that changing the Mid Cap improves things quite dramatically - a 0.01 in parallel with the stock 0.01 gives a total of 0.02 (Fender Bassman type value). This works well with a Strat but makes my Tele OD sound rather thin so I plan to make it switchable as others gave done.
Finally I tried a few OD pedals (stacked, into the clean channel of the OTS or set very low and pumped into the amps OD) - none have been particularly successful in getting a lead sound