Seems like a stretch to me. Fans don't expect him to start it is common knowledge that he is the 3rd string QB and what his abilities are. The only way people will start chanting for him to start is if the Eagles are doing terribly and Bradford and/or Sanchez are to blame.
Lets also be honest though it isn't like Tebow is joining a modest team or city that wasn't already a circus. Chip Kelly alone already drew a lot of attention and they had plenty of negative attention with Riley Cooper. In addition before the Kelly area this is the same city that brought Vick in, had the TO during his driveway sit ups phase, and of course the "Dream Team." Plus given what Kelly just did this offseason the Eagles were already going to get a frenzy of media attention this season with or without Tebow. Not to mention Philadelphia is already a big market team that naturally is used to media attention similar to the Jets and Pats who also did not have a problem with it.
I don't know why people make it seem like he is such a cancer ether. Unlike many issues other players bring it isn't like the team will have to address his bar fights, line snorting, and that fan he beat up, etc. He is a great character guy who is great with charity and community work (something he already said he wants to be "ingrained in" while in Philadelphia). It just seems overblown that him being Christian is such a horrible thing that threatens the very stability of a team xD
And lastly he signed for 1 year at around 600K or something with no money guaranteed. If they don't want him they can cut him at any time. They gave up nothing to get him and it isn't like they will look back and say boy we screwed up in bumping 3rd string QB Matt Barkley for this guy.
As far as Tebow himself goes with his Eagles role.
http://www.bleedinggreennation.com/2015/4/21/8460033/eagles-tim-tebow-quarterback-short-yardage-package-two-point-conversion-specialist
Keep in mind he had most of his success in late quarters as a Bronco which was often credited to his athletic ability being vastly superior to most others and defenses starting to tire down. The Eagle offense itself is almost built exclusively on draining the defenses energy. In addition they are a run heavy team and can feature Tebow in with both a combo of Murray, Matthews, and/or Sproles keeping plays to higher percentages and giving a diverse range.
Long story short Tebow offers more than Barkley even if it is in a role position.