If I understand, you're proposing bolting a board (say 1-1/2" to 2" thick) to the floor, then toe-nailing (or pocket-screwing most likely) what you all are referring to as "pilasters" into this board -- then yes that would work as an attachment means. You could also skip the full lenth board and just bolt some wooden attachment blocks to the bottom of each side piece ("pilaster"), then bolt that to the floor. I would make this block close to full length so your 2 (or 3) bolts can be as close to the front and rear as possible for maximum stability and resistance to torquing. You really need to just bolt each of the side pieces ("pilasters") to the floor, a full-length board isn't really necessary since the rear rail does not need to be tied to the floor (and ideally should not be IMHO).
As mentioned in my earlier post, I would also suggest taking advantage of the fact that the side pieces/pilasters extend from rail to rail and either nail, screw, or dowel them to the top and bottom rails to help ensure that the rails never again seperate from the stiles. And, of course, get rid of the angle braces along the bottom rail that were responsible for tearing the piece apart in the first place.
PS - If you don't (or can't) fully disassemble these kneelers to reglue the M&T joints, invest in a glue syringe (you can buy a really nice -- and reusable -- syringe from Woodcraft) with a 1/32" tip. You can then drill some 1/16" holes that intersect each mortise in several locations and then inject the glue and tap the joint tight with a mallet -- clamp if needed/possible. Between joints, immerse the syringe tip in a cup of water to keep the [water based] glue from curing and clogging the tip. This technique works reasonably well when presented with joints that need regluing but which can not practically be disassembled first.
Hope this helps.