I mean, IF there were two tire suppliers and with limited in-season testing,
would it be in the best interest of both to develop tires that would work
with all teams and at all tracks?
Depends on how much the bigger teams offer to skew things in their favour. If you get Red Bull, Sauber and Caterham using Kumho tyres and everyone else using Pirelli, Red Bull could pay Kumho to develop the tyres to their demands and there is nothing that Sauber or Caterham could do to counter that.
If not that, then for pity's sake, the teams should be allowed to choose
which ever tires they want to run!! Why should Pirelli dictate that??
Because the FIA requires two dry-weather compounds per race and Pirelli have to pick the two that they think are most suited to the track and conditions.
You're probably right; but, it's completely frustrating that Pirelli determines
which tires will be used at which tracks and will even indicate how many laps
is the maximum the tires will last.
See above for the first part. The recommendation for maximum laps is because the FIA wanted tyres that degrade quickly and Pirelli want to let their expectations be known beforehand, so people can't turn around and say that they don't last long enough. Not that it helps, really.
BUT, we do NOT have a "competent" supplier at the moment, if they can't
get their tires "right" and have to change them mid-season...
The FIA (and to an extent, the teams) are also to blame here, not just Pirelli. Due to paranoia from teams and regulations set by the FIA, Pirelli get very few opportunities to test with an up-to-date car. What data they can get from the pre-season tests isn't as useful as doing in-season testing because temperatures are 10-20 degrees below those that would be seen in race conditions, so performance of the rubber will be different because it is outside of its operating window. More, appropriate testing of the tyres, both during their development time in the season before and throughout the season they are used would certainly help to ensure any problems with the tyres are dealt with more quickly. Pirelli haven't done a great job, but to call them incompetent is going too far, especially when there are other factors involved.
If the FIA asked for racing tyres instead of entertainment tyres, this would be less of a problem. Pirelli can produce tyres that will last as long as the Bridgestone ones did, but first they need to be asked to do that.