It's a lovely idea and something I've whimsically thought about doing in my own country.
However, often the logistics are too hard. When I tried to work it out I could figure most of it except for riding every day the horse is going to need supplementary feeding likely daily to keep up with the energy they're expending, so you'll need somewhere to get feed each day. It's one thing carrying your own food and sometimes water, but you'll also need to carry a share for the horse. This usually means at least two horses (one to ride and one pack horse) but again, more supplies needed.
A "drop point" system could work, where you drive around leaving supplies in areas between towns, and doing a shorter trips going through parks and small towns where there are facilities.