Definitely to make the perimeter safe against frost is better, and there's no alternative way (except maybe for very impermeable soils) to ensure that frost won't cause a problem. Any restrain induced by the perimeter can be safely reduced to acceptable behaviour through reinforcement. At gates, it is of course safer to have elements of big stiffness.
This said, I have seen a number of buildings without going to frost depth and behaving well. I particularly remember one with structure in reinforced concrete, floor level plus 2 stories, where a 35 cm thick mat was placed upon 15 cm of stone and then quite malleable clay. I think to remember 45 cm square perimeter ties -and maybe between some columns- were used there. This 20 year old building has not given a problem, yet has not the required depth to be safe against frost, at least not the customary 90 cm or better 120 cm used here to such purpose (which may be an excessive requirement, anyway, since daily average temperatures almost never stay under zero ºC in this area, or just a few days).
Of course if you are thinking of a thin slab on grade you may end needing some thickening around columns, maybe even at your small loads. However, within the limits of putting in place what required (that is different in the mind of every designer) we have come to the conclusion that using single thickness is by far the cheapest and soundest way to proceed (this usually referred to foundations under 1 cellar) as soon as a mat or a piled mat starts to look what required, which has been in the tens of times in our case, given gypsum layers underneath being quite common in our area.