I notice that this issue recurs about every year. To reiterate several previous posts, we are using Sunday Streams and have been doing so for the last 3 years. In that time we've enjoyed 1080P webcasts with no outages or even stutters. The one anomaly we thought was due to Sunday Streams was actually a camera that was double framing during a pan (since fixed).
We made this move to get rid of the laptop at the receive end and use Roku boxes instead (AppleTV and Rpi work also). We found that the laptop was implicated in over half of all our problems. Since this move, webcasts have become trivial for us with literally 20 minute setup of cameras and 5 minutes at remote locations.
Sunday Streams allows payment for the month of use, so it's an annual cost of $100. Well worth the stability we've found. In addition, they are password protected, do not maintain a recording, do not require reservations (just a monthly bandwidth limit which we've never come close to), and have excellent customer support.
I look forward to having a similar system supported by the Church webcast group, whom I've talked with several times. I really hope we get there some day.