I don't like to strip blots at all. If my western doesn't work and my positive control stays blanc, I prefer to repeat everything and start a technical replicate with a new SDS-PAGE. If it doesn't work but my positive control is fine, then I do a biological replicate, and if I get the same result, I trust my technical skills and decide that my experiment doesn't yield detectable gene products with my antibody.
When I want to probe with two different antibodies and compare the bands, I prefer to run a big gel with two sets of identical samples, blot them together (molecular weight marker is in between the two sets and is cut in half), and incubate one set with one antibody and the other set with the other antibody. And then I'll do a technical replicate of both (new SDS-PAGE and everything) to see how reproducible it is.
I am using ReBlot Plus Strong Antibody Stripping Solution, 10x from Miliipore. I have to strip my blot at least twice everytime and it is working prefectly. I haven't seen much loss of proteins either.
We use Antibody stripping buffer from Euromedex. It is relatively mild, cheap, and without Mercapto. Best is to start with your protein of interest and do the strongly expressed Housekeeping's afterwards. Or as mentioned above, if you have tons of material, load several lanes and cut.