There is NO good solution to this dilemma.
If the USGA allows those Thursday tickets to be used on Friday or Saturday, that is unfair to the other fans that already have tickets to only one of those rounds. Now instead of trying to watch the Open with 50,000 other fans, they may be trying to watch the Open with 65,000 or more other fans. That means less available parking, longer lines, less available bleacher seating, etc.
If an MLB game is rained out, they do not allow that ticket holder access to any future game; they are only granted access to a non-soldout game.
Although, many tickets are sold for the whole week, the tickets are not distributed as a single weekly pass; rather 7 individual tickets (one for each day) are sold. That means that suddenly everyone that bought a weekly package, can now still see the next round, and can now also bring a friend to use the other ticket.
The USGA used to only sell weekly tickets, but since 2002 (at least) they have also been selling one day tickets. With the slower sales for this year’s Open, I would expect that more single day tickets may have been sold than any time in the past.
Although the Monday option is not the perfect solution, it may be the least harmful solution to all other ticket holders and to the logistical capabilities of this tournament.
Granted, some of those possessing tickets only to Thursday's round can not come out to the tournament on Monday, but they can still give or sell those tickets to someone else.
This is a risk to all ticket purchases for events exposed to weather. During my upcoming summer vacation I have tickets for an outdoor musical show, and a baseball game. Ironically each ticket is the final night that I can see each event during my vacation. Although each event allows future use of those tickets if a rainout may occur, neither one offers any refunds. This is a very common practice. I knew the risk and I accepted the risk.
I too felt the original USGA stance was very poor PR, but I am very impressed that they have listened to the negative feedback and offered a new option to those disappointed fans.