There have been a lot of posts over the years about batteries failing after 4 years with the gen3 MDX; I would consider that normal, depending on usage. You could take it to a shop and have them do a proper test but then you would be obligated to pay them to replace it. Alternatively just go out and buy a quality battery and DIY. Personally I am a low-mileage driver these days and my battery is still working on my 2016, delivered in July, 2015. I know I am on borrowed time. Also it seems typical that when the battery fails it does so as a complete surprise - no warning, no struggling to start the car. It just doesn't start the car, after having worked 100%. I am carrying a lithium jump-start kit just in case, which I have used to start several cars and trucks with big engines - no problem - and a boat with 2 rotax engines (difficult, but there were other problems than just a weak battery). It is a good investment.
As for the fob, the standard fob has a low frequency receiver and a long range VHF (?) transmitter. The car is constantly transmitting a low power, low frequency identification. If the fob receives that (it is constantly listening), which usually requires being within a meter or so of the car, it responds with its matching code. That is how the car knows to let you operate the various door and deck switches, as well as the ignition button. The remote start kit probably adds a long range transmitter to the car and a receiver to the fob. Considering the short range of the car's standard transmitter I doubt that the power draw is significant. The standard fob will need a new battery every 2 years or so - keep a spare in the glove box (I put one in the clear window for business cards in the car's wallet). If the fob battery is too weak to open the door then one can use the physical key to open, then replace the fob battery and go - there is no hardware lock for starting.