His approach to football is actually very similar to Mourinho's, and they actually share a lot in common, they are undoubtedly both pragmatists, they are a little bit better than people give them credit for when it comes to getting the best out of top players, yeah they are defensive minded, both are a little brash, but both are quality when it comes to digging out results, and both get their fair share of criticism when it comes to performances. They are actually very good friends too.
The biggest difference really is the calibre of clubs they have managed. Mourinho took over Porto, won the CL, you have to give him amazing credit for that, but the draw in the knock out stages opened up for him massively that year, he played a poor Man United team in the first knock out game, and got through with a late winner, with Mourinho famously running down the touchline, then after some freak results in other games that year, it meant he got to play Deportivo La Coruna, Lyon & Monaco en route to winning. Fair enough he still had to beat those teams, and he did, but were it not for that run he might not be where he is today.
I'm not trying to diminish Mourinho's achievements here, but sometimes in football you need a bit of luck, and you need to be in the right place at the right time, and after Mourinho won the CL for Porto, he won the oil lottery with Chelsea, then took over a dominant Inter team which had some world class defenders, and guys like Sneijder and Eto'o, again he done great to win the CL again, then he got the Madrid job, and now has the Man United gig.
Allardyce has simply never had those opportunities, the biggest opportunity he has had before Everton was Newcastle, and they kicked him out after 19 games or something, so they hardly really gave him a chance. But he has performed miracles in his own way, just not at the same calibre of clubs. And has over-achieved almost everywhere (when given a bit of time).
And actually, all those clubs he managed found life after Big Sam very difficult. Most of them getting relegated within a year or two of him leaving (Blackburn, Newcastle, Bolton, Sunderland, Palace struggling this year). The only club that pushed on without him were West Ham, but even that only lasted a year, now they are back to struggling again.
So you may well already have a Mourinho calibre manager on your hands, this is the biggest club he has managed at, he just needs the opportunity, and with a pre-season under his belt and a bit of faith he could take you places.
I think we are guilty sometimes of overlooking what he has achieved, because he's English, because he is a bit shall we say overweight, because he is linked with struggling clubs and playing long ball (which is a lazy generalisation) we think "oh it is only big Sam", but he brings a wealth of experience with him and has over achieved almost everywhere he has gone.
He's probably destined for the exit door at the end of the season anyway, difficult for anyone to do their job properly with so much negativity, but in case he does stay I think try and support him, give the guy a break and you might be pleasantly surprised with what he can do next season.