DVD Shrink is primarily a compression program for reducing the size of 8.5 GB double-layer DVDs down to 4.7 GB. It’s not strictly-speaking an authoring program and its burning facilities take advantage of other installed programs such as DVD Decrypter. You can’t open an MPEG2 (.mpg) file with it.
Power DVD is a player only, as you say.
Ulead DVD Movie Factory is a cheap and cheerful authoring program, or you could try the authoring version of TMPGEnc which many others would advise. If you want to go the freeware route, don’t mind not having fancy menus and have some good PC skills, then you can demultiplex the MPEG2 file to m2v (video) and mp2 (audio) streams and use IFOedit to author a DVD-Video.
What is the make and model number of your home DVD player? Some of these will play bare MPEG2 files without authoring (if they’re less than 2 GB).