It's definitely an option. It will do the things that you want (as long as your phone is online, but that is the same for any other solution).
sending Signal messages with it would be less secure
Yes, this is because Beeper converts the Signal protocol to the Matrix protocol and vice versa. In order to do this it needs to access the messages. So it needs to decrypt the messages, then re-encrypt them on the other side. This means that the bridge (in this case operated by Beeper) has access to your messages. This is often referred to as "end-to-bridge" encryption, as it isn't end-to-end anymore.
This is going to be true of any bridge you use that is hosted by a third party. You are always adding one additional trusted party into your communication.
the recommended bridge instructions sends me over to Beeper, since I don’t have my own server
Yes, to practically operate a bridge you need your own Matrix server. This is because the bridge will create a new Matrix user for every remote participant (every phone number you communicate with in this case). Doing this with regular mechanisms would be difficult (as signup is likely restricted in some ways) and inefficient (as each account would need to be checked for new messages separately). Beeper runs their own homeserver so that they can operate their bridges. However Beeper's bridges are only available to users on the same homeserver (this is not a protocol limitation, just their choice). So in order to use their bridges you need to make an account with them (which you can, it is free IIUC). Beeper also offers custom clients which have special features for interacting with their bridges (for example making it easier to start a conversation with a new phone number).
The alternative would be to run your own server and bridge (or hire someone to it on your behalf).