A close reading of the 1e AD&D rules finds that it was always intended that your armor class was composed of two things - your armor class (AC) and your bonus to your armor class (hereby called AB). To make the weapon vs. AC tables work, you have to start strictly separating AC from AB and recording the two separately - which 3e D&D pretty much does and in fact makes use of in its own way (touch AC versus flat-footed AC, for example). Unfortunately, the 1e MM got published I think before Gygax really worked out this idea, but you can see hints of it even in the MM where Gygax mentions that the monsters hide should be equivalent to a certain type of armor. That detail is only really useful if you are thinking about AC being made up of two different things. So step one of integrating the weapon vs. AC tables is every time you use a monster, make a note of what portion of their AC you think is strictly AC and what part is strictly AB. So you might have AC: 10 (+4) and AC: 4 (-2) and those are going to be from here on out different than AC 6 (+0).
Now that you've set up the game to use the table, the second thing you have to do to make it playable is precompute a PC's to hit AC tables using their class, weapon, and standard modifiers (strength bonus, magical bonus, etc.). Write this down for every PC and weapon they typically use and pin it to your DM screen.
What I discovered is that after I did that second step, my games actually sped up rather than slowed down. Players could just flat report the number that they rolled on a D20 to hit, and they wouldn't waste time doing the math in their head to do something like 14+2+3 = 19 every single attack. It saved seconds on every attack roll and that added up big time over the course of a combat and a session.
It was a slight effort to work up tables for what a monster needed to hit the PCs, but you could do it once on the fly with just a few moments of setting up or if you knew the encounter was going to happen ahead of time you could work it out in prep, or in a pinch if you were in a rush you could just assume the monsters natural weaponry was such that it had no modifiers versus AC. I've never been one to sweat +1/-1 either way because there is a small chance it even comes up, and remember even if you do quickly precompute those numbers this is still going to be more efficient than looking it up in the to hit table every roll.