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The Robins were already 1-0 down and heading for their fourth home league loss on the spin when Branco was shown a straight red card for a petulant and needless elbow on Bolton's Lawrie Wilson.
It now means Branco will miss the next three League One matches for Swindon while Nathan Thompson is currently the only available recognised centre back as both Jamie Sendles-White and Lloyd Jones are nursing injuries.
"(The loss) is very hard to take", Embleton said at the final whistle.
"It is damaging, certainly. I don't want to go down the lines of saying: 'the wheels are coming off' type stuff at the moment.
"It was not a convincing performance. Collectively I thought we defended the aerial threat that we knew we were going to get quite well.
"Throughout the game they had one or two moments where Lawrence (Vigouroux) kept us in it but I felt we lacked that threat at the top end of the pitch."
Embleton faced the media on Saturday afternoon due to manager Luke Williams being in a meeting with chairman Lee Power.
"With Raphael you can see the frustration and it shows that he has let it get to him, for want of a better phrase.
"He is foolishly going to be missing a number of games."
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