Shouldn't it be closed off.
Opinions, everybody has one, not all are the same. What are your goals?
The small, open tip drain is the later style which started sometime in the year of s3 production. It works fine as is, at least it did with the Solex carb on the 1725 in the Husky. Obviously, the factory considered it to be a needed item or the money for the machining, parts, and installation labor involved would NOT have been spent.
Things change over the course of time. Your Weber is not the same as the OE Solex. Today's fuel mixtures are not the same as was available when these engines were designed and specced in the 60's. Even if you are using non-ethanol fuel, today's fuel generally has a more volitale compound combination to aid vaporization as an emissions improvement aid. If you are using a fuel with ethanol, then due to the difference of BTU's contained in gasoline and ethanol at a 90/10 ratio, you are only delivering about 95% of the fuel BTU's the engine thinks it should have at a given load and throttle position. In other words, you are 5% lean at all times if your intake system is delivering the fuel volume of your engine's OE design parameters.
Thoughts and opinions on improving your hesitation situation:
Your drain - for now - leave it in place. For tuning purposes, you could hook a vacuum gauge there.
I used to wrap the lower part against the exhaust down-pipe - open tip away from down-pipe. When the engine is running, the overall amount of air delivered to the manifold is a small percentage of requirements. With vacuum in the manifold, it's not possible for fuel to run out. By heating the drain pipe, the air traveling through the pipe is heated which also lowers the volume of air ingested through the drain. When the engine is shutdown relieving manifold vacuum, IF there is fuel puddled on the manifold floor, that fuel if it runs down the drain will boil and rise as vapors back into the manifold until some cooling has occurred.
The point is the drain is not a significant part of your hesitation issue. Use it or not as suits your opinion.
Now for actual suggestions of possible benefit. The generic Weber 32/36 DVG used on the OE manifolds is a "best guess" for Weber's engineers as far as jetting in an installation. It's a compromise to be a "close" a fit for as many applications as possible. It's not fond of the log style manifold. It was found long ago that some Alpines had a hesitation on initial throttle input. Sometimes that can be helped to some degree by enriching the idle jet, possibly the primary main jet, and maybe the pump shot timing. Sometimes that isn't very effective.
Before playing with the carb jetting, checking the ignition system would be a higher priority. Pully timing mark lines up with TDC, no major wear in distributor shaft, vacuum advance canister hooked up properly and working ( if you've not tried running an engine with a disconnected or defective vacuum advance, it causes a bad off idle response and destroys fuel economy ). In short, verify everything in the ignition system is optimized, adjusted properly, and correctly functioning. You'd be surprised how many carbs are blamed for a challenged ignition system.
Finally, if you have a PCV valve connected, try blocking it off and see if re-tuning the engine shows any improvement. If you have the OE PCV then it may be worn out, missing parts, or dirty. If your OE PCV has been replaced with some similar valve that fits, it might be one that is designed to have the vacuum source applied in the other direction. The amount of flow is governed with manifold vacuum pulling the spring loaded spindle against its seat to restrict flow at high vacuum. If the valve is flowing in the wrong direction, there is a chance the spindle's retainer can be broken or defeated allowing the engine to ingest the spindle and destroying the engine - it has happened.
Sorry for the dissertation but maybe it can help,