Yes there is a solution. The reason it took you days to resync was because you ran the reference implementation, Bitcoin Core, which is meant for "full nodes" running 24/7. Simple users like you should instead run "thin client" or SPV wallets (Simple Payment Verification) which do not need to store the whole block chain, allowing your wallet to resync pretty much instantly. For a list of SPV wallets see
https://bitcoin.org/en/choose-your-wallet and look for those labelled "simplified validation".
You also very likely ran into the problem that older versions of Bitcoin Core were slow to download blocks. For example version 0.10.0 released in February 2015 introduces parallel block download, which makes resyncing faster. Numerous other optimizations were made over the last 12-24 months (eg. better detection of stalled downloads). From your description it sounds like you were running a significantly older version of Bitcoin Core, so you were missing out on all these optimizations.
I would like to point out that despite the block chain growing continuously, we are still very far from the point where only large service providers can hold the block chain. The average block size is currently ~400 kB (https://blockchain.info/charts/avg-block-size?showDataPoints...), this means the block chain grows by ~20 GB every year. This is only 1% the size of an average desktop hard drive (2 TB), and would take only 4.5 hours to download (once a year, mind you) assuming a steady common residential Internet download speed of 10 Mbit/s.
Heck, a dedicated computer on a common residential Internet connection could still run as a full node and handle an average block size 100x larger that today (~40 MB per block, 2 TB growth per year = 0.5 Mbit/s continuous download). I would hardly describe something an enthusiast can do as a thing that "only large service providers can do". And keep in mind by the time 100x growths happens, CPUs/HDDs/Internet speeds will be much faster than today. Bitcoin's scalability has already been benchmarked pretty well: https://en.bitcoin.it/wiki/Scalability - no immediate bottlenecks are in sight.
"high fees in its transactions"
Fees are optional, and typically 0.0001 BTC (3 cents).
"up to 15 minutes for the transaction to be approved"
No, transactions are typically "received" instantly within seconds, and take 10-60 minutes on average to "settle" (be confirmed by 1 to 6 blocks). This is a lot faster than credit cards or ACH which both take 24h (or more if this is a weekend) to be received (ie. dollars available in the recipient's account), or bank wires which take days (or end-of-business-day if your are lucky).