Robert made good suggestions. I will add my $0.02.
I have driven w123 cars many miles and have learned several things that may help. The first is that with alternators and starters both, it seems to be a waste of money to buy anything but Bosch certified remanufactured units. This is more so with starters, but you will be money ahead in the long run with Bosch certified alternators as well.
Also, many times the failure is the regulator/brush unit in the alternator.
I also once had the connector that plugs into the alternator fail. I was able to find the bad connection and solder it back in place, that repair has lasted over 100,000 miles. The rectangular plastic cover can be pried off the connector, the connector repaired and the cover snapped back in place. Don't pry off the cover until you determine that there is indeed a bad connection.
Also, as with any alternator, don't overtension the drive belt, it must be adequately tensioned, but too much will prematurely wear the drive end bearing. Too little tension will allow the belt slip, producing excess heat.
You may already know this, but the simplest way to see if the alternator is producing energy, is to simply take a DC voltmeter measurement across the battery with the engine running. The battery will read about 13 to 14 1/2 volts with the engine running, and only about 12 volts with the engine not running.
I hope this all helps.
'84 Euro 240D, 516K miles
'88 300E 5 Speed
Over 800,000 miles in