I have several suggestions to fix this problem
1) No loot from LFR. - Obviously not a desirable option. It is generally understood that the players who LFR is designed for don't just want to "see the content", they also want some purples to go with that content. As someone who does have the available time to raid normal modes I subjectively wish that LFR did not award purples but understand that my opinion is perhaps not what is best for the general WoW community. HOWEVER, it is possible to at least keep the gear from being awarded until several weeks/ possibly a month after the raids are released. This would prevent the weekly gear grind by raiders but would still probably see people running them for practice possibly.
2)Have LFR share raid lockout with heroic and normal modes - Again I don't know how feasible this one would be to implement, especially with the ability to join a LFR at any stage in the progression. I still think that its possible and this would completely eliminate raiders from going to LFR on their mains. However, most raiders would still probably experience the raid for the FIRST time in LFR, before their guild has gotten their raid group completely together, and then would never go back to it. (obviously the top of the line guilds would ignore LFR in almost all scenarios) This also has the problem of still being an alt gear up fest and perhaps ironically could end up with your alt better geared than your main depending on your dice luck.
3)Make LFR hard - not normal mode hard obviously, but the current difficulty is laughable. I have not yet wiped and have run it multiple times with some of the worst players I have ever seen take part in a raid. Now there is some inherent difficulty baked in to the LFR model of anonymity. Taking those same 25 players and adding ventrilo and the accountablity and social ties of a guild would skyrocket their capability. But there is still a tuning that would mean that only the LFRs that are filled with semi-competent people could clear the WHOLE instance. I firmly believe (though perhaps I am wrong) that the casual players who LFR is designed for still want a challenge and that the current tuning of LFR does not provide that. Again, this wouldn't really solve the problem but it would at least make running LFR more tolerable and better paced..
4)Release LFR later than normal and heroic modes!-This is i believe the best way to find a harmonic place for LFR in the WoW raiding model. Even if LFR is released at the same time as the normal and heroic mode raids, the people who will have leveled fast enough and geared up fast enough to join it will undoubtedly be the raiders themselves, not the intended demographic for LFR, and almost certainly there to get the gear and get that leg up for the regular raids. If we delay LFR's release for several weeks or possibly a month or two after the regular raids are starting up, hopefully raiders will be already comfortably progressing through normals and will only use LFR when they are bored or perhaps are behind in the gear department. This enables the raiders who want to feel that pacing exclusively in the challenging raids to ignore LFR, and the casual players who LFR was designed for will still be able to clear the content while it is relevant (towards the end of when its relevant but still)
To sum this all up, LFR should NOT feel like a stepping stone for middle of the road RAIDERS to the regular mode raiding, and it should DEFINATELY not feel like a requirement in terms of gear to run every week during the first weeks of raid progression. This would destroy the pacing of these instances that I honestly almost always love that blizzard has designed. By delaying the release of LFR, i believe that blizzard would preserve the content's availability to those it is meant for, without burdening and burning out raiders.