In my opinion if your wood plane should be 'heavier' you are using the wrong plane. Get one of the correct weight or use a steel plane. Wood planes are not all made of beech though many cabinet maker's jointing planes and moulding planes' are. Some use heavier timber. Stability is what is needed and neither timber nor planes need to be dripping or even loaded with linseed oil.
My father was a "world class+" cabinet maker....I never saw him soak a plane in Linseed ....and for good reason. You don't want the oil being transferred onto the wood you are planing, You don't want distortion from quantum oil here only little there. Timber with of course always react to humidity but quality planes are well dried and perhaps quarter-sawn. as are all mine. Under usual circumstances a light bees-wax is all needed, then polished-off.
I've just found 3 x 20-22 inch wood planes in my car affected by water in out massive recent Qld. down-pours... I hadn't realised. They were in a sggy cardboard box inside another cardboard box and with a very wet towel sitting on them. Mould had started in the beds.
I dried them naturally inside my house. I could see they were water affected....Stripped them. Next, finger and then paper-cloth wiped them with linseed which has now (20 minutes later) soaked-in. I had to work on the rust on the iron and back iron. etc. ...done all that. The wedge and bed I lightly oiled, the sole also...
Now I will not be putting on more oil, no second doses, ....and I have to wait until the planes stabilise and then wipe and beeswax them.
The soles I will probably have to flat sand, as did my father (deceased 2003 at 93) .. ...I'll see how it goes as the soles were perfect before the water ingress. My advice also is to not run linseed like tap-water around the tote-slot..just go easy. Do NOT seal the mouth/cheek/bed area then fill with linseed or linseed-turps. Do NOT soak the plane. Do NOT keep filling the bed until the oil seeps out the toe and heel...It all sounds rational but it will cause problems. Wood planes are naturally dried timber....stable. Had mine no mould on them and a bit ??...I'd have dried cleaned and beeswaxed them. Wood-planes (from the right era or the right maker today) are made of the weight to do the job. They are not just thrown together willy-nilly and some were made by their carpenter owners...'fit for purpose'....weight for purpose. Periodically they can stand checking for fit and for completely flat sole.
The wedge wants wood to wood friction to work correctly ...go really easily. ON the toe and heel...areas of possible cracking let just a tad run-in , then just leave it..until wiping and beeswaxing. and then fine rag polishing.
The wedges I have installed to maintain shape and mating but will remove tonight and let just 'sit' . I'll put the wedge on paper during 'drying' of oil... I will not use the planes for at least a week. On the three together I used about a cubic inch of Linseed.
In France (I'm in Australia) I have one or two ancient wood planes some 3-4 foot long. The French knew well how to work timber so although I have not been there for 11 years, they should be fine....I presume they used them for planning flooring planks.