Aside from the other problems brought up, the effect of radically changing the value of c would have would be wide-ranging, from electrodynamics and quantum field theory to general relativity and cosmology, likely rendering the universe uninhabitable (at least, for us).
The problems with interstellar travel are really three distinct problems (aside from the issues of recycling resources, psychosocial dynamics, and so forth); propulsion, power generation, and thermodynamics, and they’re interlinked problems as illustrated below:
The propulsion issue is limited by momentum exchange, e.g. to go anywhere you have to balance the change in momentum of the ship by an equal and opposite transfer of momentum in the opposite direction. We do this with chemical propellants, and to a limited degree with ionized plasma, but this requires carrying reaction mass with the ship which means that most of the momentum is used in accelerating propellant mass that will only later be used, and the of course enough mass to reverse the acceleration at the end of the trip. You get better efficiency the faster you eject mass, but this also requires more energy per unit impulse, which not only means you have to be able to generate power but results in more ‘waste heat’, i.e. energy that is not used for propulsion.
We tend to think of energy as being an essentially unlimited resource since virtually all of our energy comes from the Sun (except that from terrestrial nuclear decay, e.g. fission and geothermal), but in interstellar space your entire energy supply has to be carried with you, which is yet more mass. And energy, once ‘used’, e.g. moved from an accessible ‘ordered’ state to an inaccessible ‘disordered’ (or ‘low temperature) state, cannot be renewed.
Finally, the problem that few people other than actual spacecraft and satellite designers consider is thermodynamics, i.e. the rejection of ‘waste heat’, not only from the propulsion system as noted above (although while in operation it will be by far the largest source) but all operations aboard the ship including computing, recycling, cooling, and whatever else is needed to maintain functionality and habitability. All of the unrecoverable waste heat needs to be ultimately rejected into the cosmic background, which means any large spacecraft is going to be mostly outward-facing radiator surface, which is again more mass (and inert mass that isn’t even usefully for eventual momentum transfer). The need for radiating surface is why the Space Shuttle Orbiter Vehicle had its payload bay doors always open while in orbit and is actually the limiting factor in International Space Station operations.
So, getting to Alpha Centauri or other interstellar destinations is not just a matter of getting there quickly but also being able to get there at all while maintaining some acceptable threshold of habitability and functionality.
Stranger
