So... you need to control five different TVs in five different rooms, and it sounds like you want to control them from some other room. The combination of web based setup, which I don't think is cheap, with replacing TVs when they die, which is the cheapest possible approach, doesn't seem to make sense. Also, any remote you program will need to be reprogrammed any time the TV on the market at the moment isn't the same brand as the ones previously bought... and brands do change.

If you want to control each one from the room it's in, I suggest the budget will be met by using the remotes that come with the TVs. Any amount of time during which more than one TV will be of the same brand will give you a bonus: the factory remote will work in more than one room.

Since TVs of the same brand will never be guaranteed to be available in the same brand or with the same remote codes if your replacement criterion is "one just died," which is completely random, it looks like good planning is your enemy here. You need the most flexibility, and use of factory remotes allows that.