With 8 years of structural engineering experience and another 30+ years of civil engineering experience, I agree that a good base is essential and that the source of the slopping must be determined and eliminated, but the injection method (foam or concrete) are viable and permanent. Injection would (by the very nature of how it works) fill in the voids (under pressure to lift the slab) and should be a cleaner/faster solution than hammering the slab, use of skid loader to remove debris and bring in new sub-base material, running a compactor, then pouring the new concrete.
In your case it's not being subjected to heavy loads (trucks at speed), so the slab (if properly poured and reinforced) should be structurally sound.
OTOH if you're considering tearing it out soon anyway please consider a properly slopping slab as only a partial solution to basement infiltration. A better solution might be to rip it out and install drainage lines (plastic drain "tile") to direct water away, maybe with some heavy visqueen (would need to know more about your situation to advice further). We built on heavy clay (open clay mines for brick factory nearby) with a finished basement, so we:
- Surrounded most of the house with garage and deck (barely gets wet underneath)
- Built on the high point of the lot (ground slopes away from the house on all sides)
- Footing drains flow by gravity away from the house plus have a sump pump connected as a backup
- Entire roof has 2 foot overhangs with 6 inch gutters and downspouts connected to underground drains that also gravity away
- Perimeter walls have waterproofing and pea stone backfill
- Surrounded the entire perimeter with concrete slab (in front of garage and at front door) or landscaping fabric under stone
