Why can't you finish a conversion so long as you live a significant walking distance from shul?
To start, what distance ARE they looking for? Typically, you are alright if you are living "within the eruv," assuming there is an eruv. For practical reasons (government permits and the availability of places to attach the eruv), the eruv may be bigger than the community would like. If you are close to the edges of an eruv, you should check with your rabbi whether that is "close enough" to the synagogue. A good rule of thumb is living within 1 mile of the shul. The "average" person can walk that distance in 12-20 minutes.
The problem is that conversion candidates may own a home or have a cheap (or free) rental arrangement outside that distance. What then? Quite honestly, you will probably have to move, regardless of the inconvenience, lost profits, or increased cost.
If you've faced this situation before, I know where you're coming from. When I became observant, I lived about 4.5 miles from shul, and I walked every other week or so. It took a long time, but I did it, even once in 110 degree heat! However, I was a renter, and I was eager to live closer than that distance! Quite frankly, I was not willing to walk that distance for long or frequently.
And I think that is the gist of most of the arguments: almost no rabbi realistically expects people to maintain the enthusiasm to walk long distances over the long-term. When you know it's short-term (for example, until your lease ends) or you are "fresh" to Judaism, the enthusiasm to walk long distances is still there.
Further discouragements to walking include injuries, sickness, and only mildly "extreme" weather. You are much more likely to walk to shul in the rain when it's 10 minutes than when the walk is 50 minutes. Same with snow and extreme heat. As I said above, I walked 4.5 miles in 110 degree heat in tznius dress (long skirt and long sleeves). For you Celsius users, that's slightly over 43 degrees. I don't recommend it. And all of these weather-related situations could seriously endanger your health. Also, walking distances that far should probably be cleared with your doctor, just as with any exercise regime.
In short, they are afraid you'll convert and then get tired of walking the distances, so you will choose to not attend synagogue services. Since shul attendance on Shabbat is considered an important part of Jewish observance, the rabbis don't want to set you up to fail. The conversion process (and the beit din) should prepare you to live a productive Jewish life, including a commitment to synagogue attendance.
In the long term, you have to consider that you will eventually grow older and may not be physically capable of such long walks. If you're younger than that, long distances are more likely to isolate the parents of young children. It's hard enough to get these parents to do more than the minimum shul observance (even with an eruv) without doubling or tripling the average distance traveled.
[All that said, I know of one case where a beit din required a candidate who moved 1.1 miles from shul to move again "within 1 mile." I think that was more about discouragement than any of the practical rationales we've discussed above.]
"BUT WAIT," you might say! "I'm different! I'm 22, I'm healthy, I'm a marathoner, I can do this!" In short, too bad. As I have said many times before, we all want to be "the exception to the rule." The rabbis can't know whether you will maintain this enthusiasm for walking three miles two years from now. And quite frankly, history and human nature tells them that you won't. There are also the considerations of weather, children, injuries, sickness, etc.
Of course, these same arguments apply to living in very high floors of apartment buildings. I kid you not, I know people who live(d) on the 14th and 30th floor. Imagine walking that every Shabbat!